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%From: barnett@wuecon.wustl.edu
%Date: Wed, 24 Sep 97 17:45:49 CDT


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\begin{center}
{\bf NONLINEAR\ AND\ COMPLEX\ DYNAMICS\ IN\ ECONOMICS}
\end{center}

\bigskip \bigskip

\begin{center}
by\medskip

WILLIAM A. BARNETT

Department of Economics, Campus Box 1208

Washington University

One Brookings Drive

St. Louis, Missouri 63130-4810

U.S.A.

\bigskip

ALFREDO\ MEDIO

Department of Economics

The University of Venice

Ca' Foscari, 30123 Venice

ITALY

\bigskip

and\medskip

APOSTOLOS\ SERLETIS$^{*}$

Department of Economics

The University of Calgary

Calgary, Alberta T2N 1N4

CANADA
\end{center}

\bigskip \bigskip \bigskip \bigskip \bigskip \bigskip

$^{*}$We thank Demitre Serletis for research assistance.\newpage 

\section{Introduction}

According to an unsophisticated but perhaps still prevailing view, the
output of deterministic dynamical systems can in principle be predicted
exactly and - assuming that the model representing the real system is
correct - errors in prediction will be of the same order of errors in
observation and measurement of the variables. On the contrary - so the story
runs - random processes describe systems of irreducible complexity owing to
the presence of an indefinitely large number of degrees of freedom, whose
behavior can only be predicted in probabilistic terms.

This simplifying view was completely upset by the discovery of chaos, i.e.,
deterministic systems with stochastic behavior. It is now well known that
perfectly deterministic systems (i.e., systems with no stochastic
components) of low dimensions (i.e., with a small number of state variables)
and with simple nonlinearities (i.e., a single quadratic function) can have
stochastic behavior. This means that for chaotic systems, if the
measurements that determine their states are only finitely precise - and
this must be the case for any concrete, physically meaningful system - the
observed outcome may be as random as that of the spinning wheel of a
roulette and essentially unpredictable. The discovery that such systems
exist and are indeed ubiquitous has brought about a profound
re-consideration of the issue of randomness.

It is not difficult to understand why these theoretical findings have
captured the imagination of many economists. Since many important topics in
economics are typically formalized by means of systems of ordinary
differential or difference equations, these findings alone should be
sufficient to motivate economists' broad interests in chaos theory. But
there exists a question, or rather a group of questions, in economics,
usually labelled `business cycles', for which the field of mathematical
research under discussion is eminently important\footnote{%
There exist, of course, other areas of research in economics for which chaos
theory is, or could be shown to be, very important, i.e., technical
progress. We believe, however, that the case of business cycles can best
illustrate the role of nonlinear dynamical analysis in general and of chaos
theory in particular, especially when we look at it in a historical
perspective.}.

A scanty observation of the time series of most variables of economic
interest, such as the price of an individual commodity or the exchange rate
between two currencies, shows the presence of bounded and more or less
regular fluctuations, with or without an underlying trend. Even more
interestingly, this oscillating behavior seems also to characterize the
aggregate activity of industrialized economies, as represented by their main
economic indicators. Economists have long been concerned with the
explanation of this phenomenon. The literature on the subject is enormous
and the number of different theories equally vast. However, if we restrict
ourselves to the `mathematical' investigation of economic fluctuations, we
observe that two basic, mutually competing approaches have dominated this
area of research in modern times.

The origin of the first approach - which we shall label `econometric
approach to business cycles' - may be traced back to the seminal works of
Eugene Slutsky (1927) and Ragnar Frisch (1933) and was later developed, and
given the status of orthodoxy, by the works of the Cowles Commission in the
1940s and 1950s. The fundamental idea of the econometric approach is the
distinction between impulse and propagation mechanisms. In the typical
version of this approach, serially uncorrelated shocks affect the relevant
variables through distributed lags (the propagation mechanism), leading to
serially correlated fluctuations in the variables themselves\footnote{%
For completeness's sake, among the impulse-propagation models of the cycle,
one should distinguish between those in which random external events affect
economic `fundamentals' (essentially, tastes and technology), and those in
which those events directly change only agents' expectations. The latter
case has been extensively studied in recent years in the economic literature
under the label `sunspots'.}. As Slutsky showed, even simple linear
non-oscillatory propagation mechanisms, when excited by random,
structureless shocks, can produce output sequences which are qualitatively
similar to certain observed macroeconomic cycles.

The ability of the econometric approach to provide an explanation of
business cycles was called in question largely on the ground that explaining
fluctuations by means of random shocks amounts to a confession of ignorance.
An alternative approach - which we shall label `nonlinear disequilibrium' -
was then developed by a school of economists who, somewhat misleadingly, was
associated to the name of Keynes. The basic idea of these authors was that
instability and fluctuations are essentially due to market failures and
consequently they must be primarily explained by deterministic models, i.e.,
by models where random variables play no essential role. Classical examples
of such models can be found in the works of Nicholas Kaldor (1940), John
Hicks (1950), and Richard Goodwin (1951).

Mathematically, these models were characterized by the presence of
nonlinearity of certain basic functional relationships of the system and
lags in its reaction-mechanisms. The typical result was that, under certain
configurations of the parameters, the equilibrium of the system can loose
its stability, giving rise to a stable periodic solution (a `limit cycle'),
which was taken as an idealized description of self-sustained real
fluctuations, with each boom containing the seeds of the following slump and
vice versa. The nonlinear disequilibrium approach to the analysis of
business cycles was very popular in the forties and fifties, but its appeal
to economists seems to have declined rapidly thereafter and a recent, not
hostile textbook of macroeconomics [Olivier Blanchard and Stanley Fischer
(1987, pp.~277)] declares it ``largely disappeared''.

The reasons for the crisis of the Keynesian style of theorizing and the
associated nonlinear disequilibrium theories of the cycle are manifold, not
all of them perhaps pertaining to scientific reasoning, and a full
investigation of this interesting issue is out of the question here.
However, there exist two fundamental criticisms, raised against the
nonlinear disequilibrium approach mainly by supporters of the rational
expectations hypothesis, which are relevant to our discussion and can be
briefly summarized. First, in the nonlinear disequilibrium approach, agents'
expectations, either explicitly modelled or implicitly derived from the
overall structure of the model, are, under most circumstances, incompatible
with agents' `rational' behavior. Second, the nonlinear disequilibrium
approach has been `refuted' by empirical observation, as time series
generated by the relevant models do not agree with available data\footnote{%
For example, so the argument runs, a time series generated by a model
characterized by a stable limit cycle will have a power spectrum exhibiting
a sharp peak corresponding to the dominant frequency, plus perhaps a few
minor peaks corresponding to subharmonics. On the other hand, aggregate time
series of actual variables would typically have a broad band power spectrum,
often with a predominance of low frequencies - see, for example, Clive
Granger and Paul Newbold (1977).}.

The first of these criticisms can be best appreciated by making reference to
the original formulation of the rational expectations hypothesis. In John
Muth's (1961, pp. 316) own words, ``the expectations of firms (or, more
generally, the subjective probability distribution of outcomes) tend to be
distributed, for the same information set, about the prediction of the
theory (or the `objective' probability distributions of outcomes)''. If
expectations were not rational in the sense defined above -- so Muth's
argument continues -- ``there would be opportunities for economists in
commodity speculation, running a firm, or selling the information to the
present owners''\footnote{{{\it Ibid.,} pp.~330.}}. This argument does have
some validity if the outcomes of the dynamical system under consideration
can be accurately predicted once the `true' model is known, i.e., if the
outcome is periodic. In this case, if agents are `rational', fluctuations
can be explained only by exogenous random shocks\footnote{{Even when the
outcome is periodic, however, if the periodicity is long and the time path
very complicated, one may question the idea that well-informed real economic
agents would actually forecast it correctly: all the more so if the outcome
is quasi-periodic, possibly with a large number of incommensurable
frequencies.}}.

However, if the theory implies chaotic, unpredictable dynamics of the
system, the rational expectations argument loses much of its strength and
non-optimizing rules of behavior -- such as adaptive reaction mechanisms of
the kind assumed by the nonlinear disequilibrium models, or `bounded
rationality' {\it \`{a} la} Simon -- might not be as irrational as they may
seem at first sight. At any rate, the presence of chaos makes the hypothesis
of costless information and the infinitely powerful learning (and
calculating) ability of economic agents, implicit in the perfect
foresight--rational expectations hypothesis, much harder to accept.

Equally strong reservations can be raised in relation to the second
criticism mentioned above. It is well known, for example, that deterministic
chaotic systems can generate output qualitatively similar to the actual
economic time series. However, none of these broad considerations can be
used as a conclusive argument that business fluctuations are actually the
output of chaotic deterministic systems. They do, however, strongly suggest
that, in order to describe complex dynamics mathematically, one does not
necessarily have to make recourse to exogenous, unexplained shocks. The
alternative option - the deterministic description of irregular fluctuations
- provides economists with new research opportunities undoubtedly worth
exploiting.

In the following sections, we shall try to make the basic notions of
complexity, chaos, and the other related concepts more precise, having in
mind their (actual or potential) applications to economically motivated
questions. In so doing, we shall divide the presentation into two broad
parts, nicknamed `deductive' and `inductive'. The former will deal with the
analysis of given dynamical systems, whereas the `inductive' part will
consider the complementary problem of studying economic time series as
output of unknown systems.

In particular, we have four tasks before us. First, we divide the
`deductive' part of the paper into two subparts, nicknamed `geometric' and
`ergodic', and we (thus) discuss two fundamentally different approaches to
the study of dynamical systems - the geometric approach (based on the theory
of differential/difference equations) and the ergodic approach (based on the
axiomatic formulation of probability theory). Second, we discuss the
question of predictability in a rigorous manner to provide a very powerful,
but abstract way of characterizing chaotic behavior. Third, we introduce
specific applications in microeconomics, macroeconomics, and finance, and
discuss the policy relevancy of chaos. Finally, we briefly discuss several
statistical techniques devised to detect independence, nonlinearity, and
chaos in time-series data, and report the evidence of chaotic dynamics in
economics and finance.

\section{The Geometric Approach}

Chaos theory is a very technical subject and a proper understanding of the
issues at stake requires that some fundamental concepts and results be
discussed in detail.

Generally speaking, in order to generate complex dynamics a deterministic
model must have two essential properties: (i) there must be continuous- or
discrete-time lags between variables and (ii) there must be some
nonlinearity in the functional relationships of the model. In applied
disciplines including economics, the first of these features is typically
represented by means of systems of differential or difference equations and
- even though there exist other mathematical formulations of dynamics which
are interesting and economically relevant - in this paper we shall
concentrate our attention on them. The geometric (or topological) approach
to dynamics, which can be largely identified with the qualitative theory of
differential/difference equations, aims at the study of the asymptotic
geometric properties of the orbit structure of a dynamical system.

\subsection{Continuous and Discrete Dynamical Systems}

Typically, a system of ordinary differential equations will be written as%
\footnote{%
Systems described by Equation (1), in which $f$ does not depend directly on
the independent variable $t$ are called {\it autonomous}. If $f$ does depend
on $t$ directly, we shall write 
\[
\dot{x}=f(x,t),\qquad (x,t)\in {\Bbb R}^{n}\times {\Bbb R} 
\]
and $f\colon U\to {\Bbb R}^{n}$ with $U$ an open subset of ${\Bbb R}%
^{n}\times {\Bbb R}$. Equations of this type are called {\it non-autonomous}%
. In economics they are used, for example, to investigate technical progress.%
} 
\begin{equation}
\dot{x}=f(x),\qquad x\in {\Bbb R}^{n}  \label{1}
\end{equation}
where $f\colon U\to {\Bbb R}^{n}$ with $U$ an open subset of ${\Bbb R}^{n}$
and $\dot{x}\equiv dx/dt$. The vector $x$ denotes the physical (economic)
variables to be studied, or some appropriate transformations of them; $t\in 
{\Bbb R}$ indicates time. In this case, the space ${\Bbb R}^{n}$ of
dependent variables is referred to as {\it phase space} or {\it state space}%
, while ${\Bbb R}^{n}\times {\Bbb R}$ is called the {\it space of} {\it %
motions}.

Equation (1) is often referred to as a {\it vector field}, since a solution
of (1) at each point $x$ is a curve in ${\Bbb R}^{n}$, whose velocity vector
is given by $f(x)$. A solution of Equation (1) is a function 
\[
\phi :I\to {\Bbb R}^{n} 
\]
where $I$ is an interval in ${\Bbb R}$ (in economic applications, typically $%
I=[0,+\infty )$), such that $\phi $ is differentiable on $I$, $[\phi (t)]\in
U$ for all $t\in I$, and 
\[
\dot{\phi}(t)=f[\phi (t)],\qquad \ \forall \ t\in I. 
\]
The set $\{\phi (t)\mid t\in I\}$ is the {\it orbit} of $\phi $: it is
contained in the phase space; the set $\{(t,\phi (t))\mid t\in I\}$ is the 
{\it trajectory} of $\phi $: it is contained in the space of motions.
However, in applications, the terms `orbit' and `trajectory' are often used
as synonyms. If we wish to indicate the dependence on initial conditions
explicitly, then a solution of Equation (1) passing through the point $x_{0}$
at time $t_{0}$ is denoted by 
\[
\phi (t,t_{0},x_{0}), 
\]
(if $t_{0}$ is equal to zero it can be omitted). For a solution $\phi
(t,x_{0})$ to exist, continuity of $f$ is sufficient. For such a solution to
be unique, it is sufficient that $f$ be continuous and differentiable in $U$.

We can also think of solutions of ordinary differential equations in a
slightly different manner, which is now becoming prevalent in dynamical
system theory and will be very helpful for understanding some of the
concepts discussed in the following sections. Suppose we denote by $\phi
_{t}(x)$ the point in ${\Bbb R}^{n}$ reached by the system at time $t$
starting from the point $x$ at time $0$, under the action of the vector
field $f$ of Equation (1). Then the totality of solutions of (1) can be
represented by the {\it one-parameter family} of maps of the phase-space
onto itself, $\phi _{t}:{\Bbb R}^{n}\rightarrow {\Bbb R}^{n}$, which is
called {\it phase flow} or, for short, {\it flow} generated by the vector
field $f$, by analogy with fluid flow where we think of the time evolution
as a streamline.

If we now take $t$ as a fixed parameter and considering that, for autonomous
vector fields, time-translated solutions remain solutions [i.e., if $\phi
(t) $ is a solution of Equation (1), $\phi (t+\tau )$ is also a solution for
any $\tau \in {\Bbb R}$], the problem may be formulated as 
\begin{equation}
x_{t+1}=T(x_{t}),\qquad x\in {\Bbb R}^{n},\ t\in {\Bbb N}
\end{equation}
where $T=\phi _{\tau }$ and $\tau $ is the fixed value of the parameter $t$,
normalized so that $\tau =1$.

Thus, a difference equation like (2) can be derived from a differential
equation like (1). This need not be that case and many problems in economics
as well as in other areas of research give rise directly to discrete-time
dynamical systems. In fact, {\it non invertible} maps such as the celebrated
logistic map extensively discussed later in this essay could not be derived
from a system of ordinary differential equations.

Equations like (2) are often referred to as {\it \ iterated maps} since its
orbit is obtained recursively given an initial condition $x_{t}$. For
example, if we compose $T$ with itself, then we get the second iterate

\[
x_{t+2}=T\circ T(x_{t})=T^{2}(x_{t}) 
\]
and by induction on $n$ we get the $n$th iterate,

\[
x_{t+n}=T\circ T^{n-1}(x_{t})=T^{n}(x_{t}) 
\]
Hence, by the notation $T^{n}(x)$, we mean $T$ composed with itself $n-1$
times - not the $n$th derivative of $T$ or the $n$th power of $T$\footnote{%
As an example, if $T(x)=-x^{3}$, then $T^{2}(x)=T\circ
T(x)=-(-x^{3})^{3}=x^{9}$ and $T^{3}(x)=T\circ T\circ T(x)=T\circ
T^{2}(x)=-(x^{9})^{3}=-x^{27}.$}.

Notice the following difference between the orbits of continuous-time and
those of discrete-time systems: the former are continuous {\it curves} in
the state space, whereas the latter are {\it sequences of points} in space.
Also, the fact that a map is a function implies that, starting from any
given point in space, there exists only one forward orbit. If the function
is non-invertible, however, backward orbits are not defined\footnote{%
A map is invertible if and only if it is one-to-one. For example, tha map $T$%
: ${\Bbb R}\rightarrow {\Bbb R}$ defined by $T(x)=x^{2}$ is not one-to-one,
since $T(1)=1=T(-1)$. However, the map $T$: $[0,\infty )\rightarrow {\Bbb R}$
defined by $T(x)=x^{2}$ is one-to-one (and therefore invertible).}.

The short-run dynamics of individual orbits can usually be described with
sufficient accuracy by means of straightforward numerical integration of the
differential equations or iteration of the maps. In applications, however,
and specifically in economic ones, we are often concerned not with
short-term properties of individual solutions, but with the global
qualitative properties of bundles of solutions which start from certain
practically relevant subsets of initial conditions. Those global properties,
however, can only be effectively investigated in relation to trajectories
which are somehow recurrent (i.e., broadly speaking, those trajectories
which come back again and again to any region of the state space which they
once visited).

In what follows, we shall concentrate mainly on that part of the state space
of a system which corresponds to recurrent trajectories in the sense just
indicated, which will be made more precise below. Even so, a comprehensive
analysis of the global behavior of a nonlinear system may not be possible.
In this case, the best research strategy is probably a combination of
analytical and numerical investigation, the latter playing very much the
same role as experiments do in natural sciences.

\subsection{Conservative and Dissipative Systems.}

Dynamical systems, whether of a continuous or of a discrete type, can be
classified into conservative and dissipative ones. A system is said to be 
{\it conservative}, if certain physical properties of the system remain
constant in time. Formally, we may say that the flow associated with the
system given by Equation (1) preserves volumes if at all points the
so-called {\it Lie derivative} (or {\it divergence}) is zero, i.e., we have 
\[
\sum_{i=1}^{n}{\frac{{\partial f_{i}}}{{\partial x_{i}}}}=0. 
\]
Analogously, the map given by Equation (2) is said to preserve volumes in
the state space if we have at all points 
\[
\mid \det D_{x}T(x)\mid =1 
\]
where $D_{x}T(x)$ denotes the matrix of partial derivatives of $T(x)$.

An especially interesting class of conservative systems is formed by
Hamiltonian systems. A continuous-time (autonomous) system of ordinary
differential equations like (1), $\dot{x}=f(x)$, where $x=(k,q),\ k,q\in 
{\Bbb R}^{n}$ is said to be {\it Hamiltonian} if it is possible to define a
continuous function $H(k,q)\colon {\Bbb R}^{2n}\to {\Bbb R}$ - called {\it %
Hamiltonian function} - such that 
\begin{eqnarray*}
{\dot{k}} &{}&{=(\partial H/\partial q)} \\
{\dot{q}} &{}&{=-(\partial H/\partial k).}
\end{eqnarray*}
If we consider that $(d/dt)H(k,q)=(\partial H/\partial k)\dot{k}+(\partial
H/\partial q)\dot{q}=0$, we can deduce that $H(k,q)$ is constant under the
flow.

From the fact that in conservative systems volumes remain constant under the
flow (or map), we may deduce that those systems cannot have attracting
regions in the phase space, i.e., there can never be asymptotically stable
fixed points, or limit cycles, or strange attractors. Since strange
attractors (to be defined later) are the main object of our investigation
and conservative systems are relatively rare in economic applications, we
shall not pursue their general study here\footnote{%
One interesting economic example of a conservative dynamical system is the
well-known model of infinite horizon optimal growth, which can be formulated
as follows
\par
\[
\max_{\dot{k}}\int_{0}^{\infty }u(k,\dot{k})e^{-\rho t}dt 
\]
with $(k,\dot{k}){\in S\subset }{\Bbb R}^{2n}$, and $k(0)=k_{0}$. In this
formulation, $u$ is a concave utility function, $k$ denotes the capital
stock, $\dot{k}$ is net investment, $\rho \in {\Bbb R}^{+}$ is the positive
discount rate, and the set $S$ is convex and embodies the technological
restrictions.
\par
To attack this problem by means of the Pontryagin Maximal Principle, we must
first of all introduce an auxiliary vector-valued variable $q\in {\Bbb R}%
^{n} $ and define a function 
\[
H(k,q)\equiv \max_{\dot{k};(k,\dot{k})\in S}\{u(k,\dot{k})+q\dot{k}\}. 
\]
{where the variables }$q$ can be interpreted as prices of investment goods
and the {\ Hamiltonian function $H(k,q)$ can be interpreted as the (maximum)
current value of national income, evaluated in terms of utility. The}
necessary (though not sufficient) condition for maximization is that the
time evolution of $k$ and $q$ satisfies the following system of differential
equations
\par
\[
\dot{k}=\frac{\partial H}{\partial q}\qquad \text{and\qquad }\dot{q}=-\frac{%
\partial H}{\partial k}+\rho q 
\]
which can be thought of as a Hamiltonian system, plus a (linear)
perturbation given by the term $\rho q$.}.

Unlike conservative ones,{\bf \ }{\it dissipative} dynamical systems, on
which most of this essay concentrates, are characterized by contraction of
phase space volumes with increasing time. Dissipation can be formally
described by the property that divergence is negative, i.e., we have 
\[
\sum_{i=1}^{n}{\frac{{\partial f_{i}(x)}}{{\partial x_{i}}}}<0 
\]
or, in the discrete-time case, 
\[
\mid \det D_{x}T(x)\mid <1. 
\]

Because of dissipation, the dynamics of a system whose phase space is $n$%
-dimensional, will eventually be confined to a subset of dimension smaller
than $n$. Thus, in sharp contrast to the situation encountered in
conservative systems, dissipation permits one to distinguish between {\it %
transient} and {\it permanent} behavior. For dissipative systems, the latter
may be quite simple even when the number of phase space variables is very
large.

To better understand this point, think of an $n$-dimensional system of
differential equations characterized by a unique, globally asymptotically
stable equilibrium point. Clearly, for such a system, the flow will contract
any $n$-dimensional set of initial conditions to a zero-dimensional final
state, a point in ${\Bbb R}^n$. Think also of an $n$-dimensional ($n\ge 2$)
dissipative system characterized by a unique, globally stable limit cycle.
Here, too, once the transients have died out, we are left with a
one-dimensional orbit, the cycle.

The asymptotic, permanent regime of a dissipative system is the only
observable behavior, in the sense that it is not ephemeral, can be repeated
and therefore be `seen' (i.e., on the screen of a computer), and is often
easier to investigate than the overall orbit structure. Even though
transients may sometimes last for a very long time and their behavior may be
an interesting subject for investigation, for dissipative systems we shall
concentrate instead on the long-run behavior of the system, ignoring the
transient behavior associated with the start up of the system. That is, we
shall consider only the attractor (or attractors, in general) to which
trajectories from a range of initial conditions are attracted, to understand
the asymptotic properties of a dynamical system. That is, we shall
concentrate on the asymptotic properties of a dynamical system, devoting our
attention mainly to the attractors of a system, i.e., to the sets of points
to which trajectories starting from a range of initial conditions tend as
time goes by.

\subsection{Invariant and Attracting Sets}

To discuss recurrence properties of orbits of a dynamical system, we shall
start from the notion of invariant sets. Such sets play an important role in
the organization of system orbits in the state space and their investigation
is an indispensable first step in the study of the dynamics of a system.
Formally, for the discrete dynamical system given by Equation (2), we say
that the set $S\subseteq X$ is{\bf \ }{\it invariant }under the action of
the map $T$, if we have:

\begin{eqnarray*}
\phi _{t}(S) &\subseteq &S,\qquad \forall \ t\in {\Bbb R} \\
\lbrack \text{respectively, }T^{n}(S) &\subseteq &S,\qquad \forall \ n\in 
{\Bbb N]}
\end{eqnarray*}
This says specifically that as we apply the map $T$ to any point of $S$,
then we obtain yet another point of $S.$

When constructing a mathematical model of the time evolution of certain
physical, or economic variables, we often wish to impose constraints on the
set $A$ of `reasonable' values of those variables. For example, quantities
such as capital stock, consumption or relative prices should remain
positive, or at least non-negative for all times; quantities such as the
saving ratio or the ratio between factor remunerations and total income
should be between zero and one at all times, etc. In other words, we want
the `acceptable' set $A$ to be invariant. The invariance of $A$ is a
necessary (although not sufficient) condition for the validity of a
dynamical model and in particular of its, implicit or explicit, adjustment
mechanisms.

In most cases of practical interest, however, finding invariant sets is not
enough. We also wish to locate the region(s) of the state space which
ultimately capture all the orbits originating in a certain (not too small)
domain. For this purpose, we suggest the following definition.

\begin{definition}
A closed invariant set $A$ is said to be an {\it attracting set}{\bf \ }if%
{\bf \ } for every open set $V\supset A$, there exists an open neighborhood $%
U$ of $A$, such that for all $x\in U$ (except perhaps certain subsets of
Lebesgue measure zero)$,$ $T^{n}(x)\in V$ for all $n>N>0$ and $%
T^{n}(x)\rightarrow A$ for $n\rightarrow \infty .$
\end{definition}

For an attracting set we can also define the {\it basin} (or {\it domain}) 
{\it of attraction},{\bf \ }as the set of points each of which gives rise to
an orbit that is caught by the attracting set. Formally, we can define the
basin of attraction as the set $B=\bigcup_{t\leq 0}\phi _{t}(V)$ [for maps, $%
B=\bigcup_{n\leq 0}G^{n}(V)$].

The fact that a set is attracting does not mean that all its parts are
attracting too. Therefore, in order to describe the asymptotic regime of a
system, we need the stronger concept of {\it attractor}{\bf . }A desirable
property of an attractor - as distinguished from an attracting set - is{\it %
\ indecomposability,} or {\it irreducibility}. This property obtains when an
orbit, starting from any point on the attractor, as time goes by gets
arbitrarily close to any other point. Strangely enough, there is no
straightforward and universally adopted definition of attractor, and
although the properties of the simpler cases can be easily dealt with, more
complicated types of attractors present difficult conceptual problems. In an
operational, non-rigorous sense, an attractor is a set on which experimental
points generated by a map (or a flow) accumulate for large{\it \ }$t%
\footnote{%
See, for example, Eckmann and David Ruelle (1985, pp.623).}$. We shall
retain this broad, operational definition here, deferring a more
sophisticated discussion of the question of attractiveness and observability
to the part of this paper concerning the ergodic approach\footnote{%
The notion of attractiveness is intimately related to that of `stability' of
orbits. Given the vastity of the subject, we cannot deal with it in any
detail here and shall refer the reader to the relevant bibliography - for a
recent, very clear discussion of this topic, see Paul Glendinning (1994).}.

The simplest type of an attractor is a stable {\it fixed point}, or, using a
terminology more common in economics, a stable {\it equilibrium}\footnote{%
In the recent times, economists often use a notion of equilibrium somewhat
different from that of mathematicians and physicists, sometimes labelled
`dynamic' or `sequence equilibrium'. Broadly speaking, the latter implies
that certain conditions hold (in a nutshell, `all markets clear') at all
times, while the system evolves in time. As one of these author argued
elsewhere [Alfredo Medio (1992, pp. 11-12)], this representation in fact
implies the presence of two dynamic mechanisms: a short-run, often hidden,
dynamics generating a temporary equilibrium (`market clearance'), and a
long-run dynamics describing the evolution of equilibria in time, not
necessarily converging to a stationary state (fixed point).}{\bf . }%
Ascertaining the existence{\it \ }of a fixed/equilibrium point
mathematically amounts to finding the solutions of a system of algebraic
equations. In the continuous-time case $\dot{x}=f(x),$the set of equilibria
is defined by $E=\{\bar{x}\mid f(\bar{x})=0\}$, i.e., the set of values of $%
x $ such that its rate of change in time is $0$. Analogously, in the
discrete-time case $x_{t+1}=T(x_{t})$, we have $E=\{\bar{x}\mid \bar{x}-T(%
\bar{x})=0\}$, i.e., the values of $x$ which are mapped to themselves by $T.$

As an example, consider the `logistic' map

\begin{eqnarray}
x_{t+1} &=&T_{r}(x_{t})  \label{3} \\
&=&rx_{t}(1-x_{t}),\qquad x\in [0,1],\ r\in (0,4].  \nonumber
\end{eqnarray}
To find the fixed points of (3), we put $x_{t+1}=x_{t}=\overline{x}$ and
solve for $\overline{x}$, finding $\overline{x}_{1}=0$ and $\overline{x}%
_{2}=1-1/r$ - see Figure 1.

To get some idea of the importance of fixed points, in Figure 2 we plot the
phase diagram of the logistic map for different values of the {\it tuning}
(or {\it control}) {\it parameter}, $r$. Notice that the height of the phase
curve hill depends on the value $r$. For $r<1$, the only fixed point in the
interval $0\leq x\leq 1$ is $\bar{x}=0$, but for $r>1$, there are two fixed
points. Using graphical iteration (an algorithmic process of drawing
vertical and horizontal segments first to the phase curve and then to the
diagonal, $x_{t+1}=x_{t}$, which reflects it back to the curve), it is easy
to show that all trajectories for starting values in the interval $0\leq
x\leq 1$ and for $r<1$ approach the final value $\bar{x}=0.$ The point $\bar{%
x}=0$ is the attractor for those orbits and the interval $0\leq x\leq 1$ is
the basin of attraction for that attractor.

In general, we can examine the dynamical information contained in the
derivative of the map at the fixed point, $T^{\prime }(\bar{x})$. If $%
|T^{\prime }(\bar{x})|\neq 1,\bar{x}$ is called {\it hyperbolic fixed point}%
. In fact a fixed point $\bar{x}$ is {\it stable} (or {\it attracting}) if $%
|T^{\prime }(\bar{x})|<1$, {\it unstable} (or {\it repelling}) if $%
|T^{\prime }(\bar{x})|>1$, and {\it superstable} (or {\it superattractive})
if $|T^{\prime }(\bar{x})|=0$ - superstable in the sense that convergence to
the fixed point is very rapid. Fixed points whose derivatives are equal to
one in absolute value are called {\it nonhyperbolic} (or {\it neutral}) {\it %
fixed points}.

Next in the scale of complexity of invariant sets, we consider {\it stable
periodic solutions}, or{\bf \ }{\it limit cycles}{\bf . }For maps, a point $%
\bar{x}$ is a periodic point of $T$ with period $k$, if $T^{k}(\bar{x})=\bar{%
x}$ for $k>1$ and $T^{j}(\bar{x})\neq \bar{x}$ for $0<j<k $. In other words, 
$\bar{x}$ is a periodic point of $T$ with period $k$ if it is a fixed point
of $T^{k}$. In this case we say that $\bar{x}$ has period $k$ under $T$, and
the orbit is a sequence of $k$ distinct points $\{\bar{x},T(\bar{x}),\ldots
,T^{k}(\bar{x})\}$ which, under the iterated action of $T$, are repeatedly
visited by the system, always in the same order. Since all points between $%
\bar{x}$ and $T^{k}(\bar{x})$ are also period $k$ points, the resulting
sequence is known as a {\it period }$k${\it \ cycle} or alternatively a $k$-%
{\it period cycle}. Notice that $k $ is the {\it least} period - if $k=1$,
then $\bar{x}$ is a fixed point for $T\footnote{%
For example, the point $1$ lies on a 2-cycle for $T(x)=-x^{3}$, since $%
T(1)=-1$ and $T(-1)=1.$ Similarly, the point $0$ lies on a 3-cycle for $%
T(x)=-\frac{3}{2}x^{2}+\frac{5}{2}x+1$, since $T(0)=1,$ $T(1)=2$, and $%
T(2)=0.$}$.

The third basic type of attractor is called {\it quasiperiodic}{\bf .} If we
consider the motion of a dynamical system after all transients have died
out, the simplest way of looking at a quasiperiodic attractor is to describe
its dynamics as a mechanism consisting of two or more independent periodic
motions - see Robert Hilborn (1994, pp. 154-157) for a non-technical
discussion. Quasiperiodic orbits can look quite complicated, since the
motion never exactly repeats itself (hence, {\it quasi}), but the motion is
not chaotic (as it was wrongly once conjectured). Notice that quasiperiodic
dynamics have been found to occur in economically motivated dynamical models
- see, for example, Hans-Walter Lorenz (1993), Medio (1992, chapter 12), and
Medio and Giorgio Negroni (1996).

Attractors with an orbit structure more complicated than that of periodic or
quasiperiodic systems are called {\it chaotic} or{\bf \ }{\it strange
attractors}{\bf . }The strangeness of an attractor mostly refers to its
geometric characteristic of being a `fractal' set, whereas chaotic is often
referred to a dynamic property, known as `sensitive dependence on initial
conditions', or equivalently, `divergence of nearby orbits'. Notice that
strangeness, as defined by fractal dimension, and chaoticity, as defined by
sensitive dependence on initial conditions, are independent properties.
Thus, we have chaotic attractors that are not fractal and strange attractors
that are not chaotic.

As we shall see, separation of nearby orbits, or, equivalently,
amplification of errors is the basic mechanism that makes accurate
prediction of the future course of chaotic orbits impossible, except in the
short run. On the other hand, as chaotic attractor are bounded objects, the
expansion that characterizes their orbits must be accompanied by a `folding'
action that prevents them to escape to infinity. The coupling of `stretching
and folding' of orbits is the distinguishing feature of chaos and it is at
the root of both the complexity of its dynamics and the `strangeness' of its
geometry.

In dissipative systems, a chaotic attractor typically arises when the
overall contraction of volumes, which characterizes those systems, takes
place by shrinking in some directions, accompanied by (less rapid)
stretching in the others. However, one-dimensional, non-invertible maps that
generate chaotic orbits characterized by sensitive dependence on initial
conditions - such as, for example, the logistic map - pose a puzzling
problem. Strictly speaking, they are not conservative or dissipative: they
might indeed be called `anti-dissipative'. These maps only have a stretching
action and their output remains bounded due to the effect of the
(nonmonotonic) nonlinearity. We could think of these maps as limit cases of
(dissipative) two-dimensional, invertible maps with very strong contraction
in one direction - so strong that, in the limit, only one dimension is left,
along which nearby orbits separate.

In what follows, we shall discuss the `fractal' property of chaotic
attractor briefly, whereas the `sensitive dependence on initial conditions'
property of chaos will be given greater attention here and in the ergodic
section of the paper, since this property of chaos is, in our opinion, the
most relevant to economics.

\subsection{Fractal Dimension}

The term `fractal' was coined by Benoit Mandelbrot (1985) and it refers to
geometrical objects characterized by non-integral dimensions and
`self-similarity'. Intuitively, a snowflake can be taken as a natural fractal%
\footnote{%
The term {\it fractal} comes from the Latin {\it fractus} which means broken.%
}. The problem of defining measurement criteria finer than the familiar
Euclidean dimensions (length, area, volume) in order to quantify the
geometric properties of `broken' or `porous' objects was tackled by
mathematicians long before the name and properties of fractals became
popular. There now exists a rather large number of criteria for measuring
qualities that otherwise have no clear definition (such as, for example, the
degree of roughness or brokenness of an object), but we shall limit
ourselves here to discuss the simplest type concisely.

Let $S$ be a set of points in a space of Euclidean dimension $p$ (think, for
example, of the points on the real line generated by the iterations of a
one-dimensional map). We now consider certain boxes of side $\epsilon $ (or,
equivalently, certain spheres of radius $\epsilon $), and calculate the
minimum number of such cells, $N(\epsilon )$, necessary to `cover' $S$.
Then, the {\it fractal dimension} $D$ of the set $S$ will be given by the
following limit (assuming it exists)

\begin{equation}
D\equiv \lim_{\epsilon \to 0}{\frac{{\log (N(\epsilon ))}}{{\log (1/\epsilon
)}}}  \label{4}
\end{equation}

The quantity defined in Equation (4) is also called the (Kolmogorov) {\it %
capacity dimension}. It is easily seen that, for the most familiar
geometrical objects, it provides perfectly intuitive results. For example,
if $S$ consists of just one point, $N(\epsilon )=1$ and $D=0$; if it is a
segment of unit length, $N(\epsilon )=1/\epsilon $, and $D=1$; if it is a
plane of unit area, $N(\epsilon )=1/\epsilon ^{2}$ and $D=2$; finally, if $S$
is a cube of unit area, $N(\epsilon )=1/\epsilon ^{3}$ and $D=3$,~etc. That
is to say, for `regular' geometric objects, dimension $D$ does not differ
from the usual Euclidean dimension, and, in particular, $D$ is an integer.

The fractal dimension, however, is not always an integer. Let us consider
the fractal called {\it Cantor set} (or {\it Cantor dust)} - named after the
German mathematician George Cantor (1845-1918). To make a Cantor set, start
with a line segment of unit length. Remove the middle third and repeat this
process without end, each time on twice as many line segments as before. The
Cantor set is the set of points that remains, which are infinitely many but
their total length is zero. What is the fractal dimension of the Cantor set?
By making use of the notion of capacity dimension, we shall have $N(\epsilon
)=1$ for $\epsilon =1$, $N(\epsilon )=2$ for $\epsilon =1/3$, and,
generalizing, $N(\epsilon )=2^{n}$ for $\epsilon =(1/3)^{n}$. Taking the
limit for $n\to \infty $ (or, equivalently, taking the limit for $\epsilon
\to 0$), we can write 
\[
D=\lim_{\stackunder{(\epsilon \rightarrow 0)}{n\rightarrow \infty }}\frac{%
\text{log }2^{n}}{\text{log }3^{n}}\simeq 0.63 
\]
We have thus quantitatively characterized a geometric set that is more
complex than the usual Euclidean objects. Indeed the dimension of the Cantor
set is a non-integer. We might say that the Cantor dust is an object
`greater' than a point (dimension $0$) but `smaller' than a segment
(dimension $1$). It can also be verified that the Cantor set is
characterized by self-similarity.

Let us consider another fractal, namely the Sierpinski triangle - named
after the Polish mathematician Vaclav Sierpinski (1882-1969). To construct a
Sierpinski triangle, we start with an equilateral triangle of unit side
length. Connect the inner midpoints of the sides with lines and remove the
inner triangle of the four equal triangles. Repeat this process to infinity,
each time on three times as many triangles as before. At its infinite stage
of growth, when the Sierpinski triangle is complete and fully grown, it will
consist of an infinite number of triangles with a total area of zero. What
is the value of $D$ for the Sierpinski triangle? Since after $n$ stages we
are left with $N(\epsilon )=3^{n}$ triangles of side length $\epsilon
=1/2^{n},$ taking the limit for $n\rightarrow \infty $ (or, equivalently, $%
\epsilon \rightarrow 0$) we have

\[
D=\lim_{\stackunder{(\epsilon \rightarrow 0)}{n\rightarrow \infty }}\frac{%
\text{log }3^{n}}{\text{log }2^{n}}\simeq 1.584 
\]
Here, $D$ is smaller than $2$, in spite of the fact that the triangle is
embedded in two dimensions\footnote{%
Similarly, the value of $D$ for the Sierpinski gasket (a $3$-dimensional
version of the Sierpinski triangle) is $2$ and the value of $D$ for the Koch
snowflake (named after the Swedish mathematician Helge von Koch who proposed
it in 1904) is $1.261...$ -- see Hans Lauwerier (1991) for a discussion of
these and other fractal objects.}.

The concept of fractal dimension is useful in the geometric analysis of
dynamical systems, because it can be conceived of as a measure of the way
trajectories fill the phase space under the action of a flow or a map. A
non-integer fractal dimension, for example, indicates that trajectories of a
system fill up less than an integer subspace of the phase space - see Medio
(1992, chapter 7) for a non-rigorous, but intuitive discussion. Also, the
concept of fractal dimension is useful in the quantitative analysis of
chaotic attractors. For example, the dimension of the attractor of a system
[as measured by ($4$)] can be taken as an index of complexity, as indicated
by the essential dimension of the system.

\subsection{Lyapunov Exponents}

To provide a rigorous characterization, as well as a way of measuring
sensitive dependence on initial conditions, we shall now discuss a powerful
conceptual tool known as Lyapunov exponents. They provide an extremely
useful tool for characterizing the behavior of nonlinear dynamical systems.
They measure the (infinitesimal) exponential rate at which nearby orbits are
moving apart. A positive Lyapunov exponent is an operational definition of
chaotic behavior\footnote{%
Notice, however, that it is possible to have sensitive dependence on initial
conditions with orbit divergence less than exponential. In this case, no
Lyapunov exponent will be positive.}.

Although Lyapunov exponents could be discussed in a rather general
framework, we shall deal with the issue in the context of one-dimensional
maps, since they are by far the most common type of dynamical system
encountered in economic applications of chaos theory. Consider, therefore,
the map given by Equation ($2$), with $T:U\rightarrow {\Bbb R}$, $U$ being a
subset of ${\Bbb R}$. We want to describe the evolution in time of two
orbits originating from two nearby points $x_{0}$ and $x_{0}+\epsilon $
(where $\epsilon $ is the difference, assumed to be infinitesimally small,
between $x_{0}$ and $x_{0}+\epsilon $)$.$ If we apply the map function $T$, $%
n$ times to each point, the difference between the results will be related
to $\epsilon $ as follows:

\[
d_{n}=e^{n\lambda (x_{0})}\epsilon 
\]
where $d_{n}$ is the difference between the two points after they have been
iterated by the map $T$, $n$ times and $\lambda (x_{0})$ is the rate of
convergence or divergence.

Taking the logarithm of the above equation and solving for $\lambda (x_0)$
gives

\[
\lambda (x_{0})=\frac{1}{n}\log |\frac{d_{n}}{\epsilon }|. 
\]
Asymptotically, we shall have\footnote{%
Notice that $\frac{d_{n}}{\epsilon }=T^{\prime }(x_{n-1})...T^{\prime
}(x_{1})T^{\prime }(x_{0}).$}

\begin{eqnarray}
\lambda (x_{0}) &=&\lim_{n\to \infty }{\frac{1}{n}}\log |\frac{d_{n}}{%
\epsilon }|  \label{5} \\
&=&\lim_{n\to \infty }{\frac{1}{n}}\log |T^{\prime }(x_{n-1)}...T^{\prime
}(x_{1})T^{\prime }(x_{0})|  \nonumber \\
&=&\lim_{n\to \infty }{\frac{1}{n}}\sum_{j=0}^{n-1}\log |T^{\prime }(x_{j})|
\nonumber
\end{eqnarray}
The quantity $\lambda (x_{0})$ is called Lyapunov exponent. Note that the
right hand side of (5) is an average along an orbit (a time average) of the
logarithm of the derivative\footnote{%
Notice that, in general, Lyapunov exponents depend on the selected initial
conditions. We shall see later under what conditions they may be independent
of them.}. From Equation (5), the interpretation of $\lambda (x_{0})$ is
straightforward: it is the (local) asymptotic exponential rate of divergence
of nearby orbits\footnote{%
It is local, since we evaluate the rate of separation in the limit, as $%
\epsilon \rightarrow 0$. It is asymptotic, since we evaluate it in the limit
of indefinitely large number of iterations, as $n\rightarrow \infty $ -
assuming that the limit exists.}.

As an example, let

\begin{equation}
T_{\Lambda }(x)=\left\{ 
\begin{array}{llll}
2x & \text{for} & 0\leq x\leq 1/2 &  \\ 
2(1-x) & \text{for} & 1/2\leq x\leq 1 & 
\end{array}
\right.  \label{Tent}
\end{equation}
be the symmetric `tent' map. Clearly, $\lambda (x_{0})$ is not defined if $%
x_{0}$ is such that $x_{j}=T_{\Lambda }^{j}(x_{0})=1/2$ for some $j$
(because the derivative is not defined). For other points $x_{0}\in [0,1]$, $%
|T_{\Lambda }^{\prime }(x_{j})|=2$ for all $j$, so that $\lambda
(x_{0})=\log 2$.

As another example, consider the logistic map, $T_{r}(x)$, given by Equation
(3). Since $T_{r}^{\prime }(x_{j})=r(1-2x_{j})$, the Lyapunov exponent is
given by

\begin{eqnarray*}
\lambda (x_{0}) &=&\lim_{n\to \infty }{\frac{1}{n}}\sum_{j=0}^{n-1}\log
\left| r(1-2x_{j})\right| \\
&=&\log r+\lim_{n\to \infty }{\frac{1}{n}}\sum_{j=0}^{n-1}\log |1-2x_{j}|
\end{eqnarray*}
Clearly, if $x_{0}=0$ or $1$, then $\lambda (x_{0})=\log r$. For points $%
x_{0}\in (0,1)$ and for $r=4$, $\lambda (x_{0})=\log 2$.

The sign of Lyapunov exponents is especially important to classify different
types of dynamical behavior. In particular, the presence of a positive
Lyapunov exponent signals that nearby orbits diverge exponentially in the
corresponding direction. In its turn, this indicates that observation errors
will be amplified by the action of the map. We shall see in what follows
that the presence of a positive Lyapunov exponent is intimately related to
the lack of predictability of dynamical systems, and thus it is an essential
feature of chaotic behavior\footnote{%
The calculation of Lyapunov exponents in the general, multidimensional case
is more complex and cannot be discussed here in any detail.}.

\subsection{Topological Conjugacy}

Before we move on, we shall discuss a fundamental type of equivalence
relation between maps, called {\it topological conjugacy}. It plays an
important role in the study of dynamical systems, since it shows that two
apparently different systems may actually be dynamically equivalent.

\begin{definition}
Let $f:X\rightarrow X$ and $g:Y\rightarrow Y$ be two maps. A homeomorphism%
\footnote{%
A map $h:X\rightarrow Y$ is a{\it \ homeomorphism} if and only if $h$ is
continuous, one-to-one, onto, and has a continuous inverse. In this case, we
say that the domain and codomain are homeomorphic to one another.} $%
h:X\rightarrow Y$ is called a {\it topological conjugacy} (or {\it %
topological equivalence}) if $h\circ f=g\circ h$. We also say that $f$ and $%
g $ are {\it topologically conjugate} {\it by} $h$, or that $f$ and $g$ are 
{\it conjugate}.
\end{definition}

\noindent The relationship can be described pictorially as a commutative
diagram\vskip 0.1cm

\begin{eqnarray*}
&& 
\begin{array}{ccc}
x\in X & \stackrel{f}{\longrightarrow } & f(x)\in X \\ 
\left. \stackunder{}{
\begin{array}{c}
\\ 
\stackrel{h}{}
\end{array}
}\right\downarrow &  & \left\downarrow 
\begin{array}{c}
\\ 
\stackrel{h}{}
\end{array}
\right. \\ 
h(x)\in Y & \stackunder{}{\stackrel{g}{\longrightarrow }} & 
h(f(x))=g(h(x))\in Y
\end{array}
\\
&&
\end{eqnarray*}
Broadly speaking, the diagram says that if we start with an element $x\in X$
in the upper left corner and then follow the arrows in either possible
direction, we always end up at the same element $h(f(x))=g(h(x))\in Y$ in
the lower right corner. The homeomorphic property of $h$ and the condition $%
h\circ f=g\circ h$ guarantee that the dynamics of $f$ on $X$ and that of $g$
on $Y$ are essentially the same - see Clark Robinson ($1995$) for more
details.

As an example, it is easy to show that the logistic map (with $r=4$), $%
T_{4}(x),$ on $[0,1]$ and $T_{\Lambda }(x)$ on $[0,1]$ are topologically
conjugate by $h(x)=\sin ^{2}(\pi x/2)$. Clearly, $h$ transforms the unit
interval $[0,1]$ to itself in a one-to-one fashion, it is continuous, and
its inverse, $h^{-1}$, is also continuous. To prove the conjugacy, it
suffices to show that $T_{4}(x)\circ h=h\circ T_{\Lambda }(x)$.

\subsection{Transition to Chaos}

In the previous sections, we have provided a classification of attractors
and discussed the distinct properties of chaotic attractors. The relevance
of these procedures would be greatly enhanced if, in addition, we could
describe the qualitative {\it changes} in the orbit structure of the system
which take place when the control parameters are varied. In this way, we
would obtain not only a snapshot of chaotic dynamics, but also a description
of its emergence. Moreover, if we could provide a rigorous and exhaustive
classification of the ways in which complex behavior may appear, transition
to chaos could be predicted theoretically, and potentially turbulent
mechanisms could be detected in practical applications - and their
undesirable effects could be avoided by acting on the relevant parameters.

Unfortunately, the present state of the art does not permit us to define the
prerequisites of chaotic behavior with sufficient precision and generality.
In order to forecast the appearance of chaos in a dynamical system, we are
for the time being left with a limited number of theoretical predictive
criteria and a list of certain typical (but by no means exclusive) `routes
to chaos'. Typically, transition to chaos takes place through {\it %
bifurcations}. A bifurcation is an essentially nonlinear phenomenon and
describes a qualitative change in the orbit structure of a (discrete or
continuous-time) dynamical system when one or more parameter is changed.
Bifurcation theory is a vast and complex area and we shall consider it here
only incidentally.

There exist various types of routes to chaos, generated by so-called
codimension one bifurcations (that is, bifurcations depending on a single
parameter). In what follows, we shall only (briefly) deal with {\it %
period-doubling}, probably the best known route to chaos at least in the
economics literature - see, for example, William Baumol and Jess Benhabib
(1989). For a discussion of other routes to chaos (such as intermittency and
the quasiperiodic route), see Medio (1992, chapter 9).

Period-doubling takes place in both discrete and continuous-time dynamical
systems, and can be most simply described by considering the dynamics of the
logistic map, $T_{r}(x)$ given by equation (3), for different values of $r$.
If $r<1,$ the phase curve will lie entirely below the $x_{t+1}=x_{t}$ line
in the positive quadrant - see Figure 3(a) - and $\overline{x}=0$ is the
only fixed point (in fact $\overline{x}=0$ is an equilibrium for all $r$).
Figures $3$(a) and $3$(b) give the phase and state space representations of $%
T_{r}(x)$ for $r=0.6$ and $x_{0}=0.01$. Notice that the only fixed point is
at $T_{r}(x)=\overline{x}=0.$

As $r$ increases beyond $1$, $\overline{x}=0$ loses stability, but a new
(positive) fixed point, $\overline{x}=1-1/r$, appears at the intersection of
the $x_{t+1}=x_{t}$ line and the phase curve, as shown in Figure $4$(a). In
fact, for $r=2$ the fixed point $\overline{x}=1-1/r$ becomes superstable -
since $T_{2}^{\prime }(1/2)=0.$ Therefore, for $1<r<3$ there are two fixed
points: $\overline{x}=0$, which is unstable, and $\overline{x}=1-1/r$, which
is stable. From Figure $4$(b) we see that the trajectory approaches some
positive unique value (a so-called {\it single limit point}) between $0$ and 
$1$.

As $r$ goes through $r=3$ , a bifurcation called `flip' occurs and the
situation changes. The fixed point $x=1-1/r$ turns into a repeller, since $%
|T_{r}^{\prime }(x)|>1$, and a stable $2$-{\it cycle} (or an {\it orbit of
period }$2$) is born: $x,T_{r}(x),T_{r}^{2}(x)=x$. For example, for $%
r=3.2360679775$, there is a superstable orbit of period $2$: $0.5$, $%
0.8090169943...$, $0.5$ - see the state diagram in Figure $5$(b).

Let us briefly describe how this happens. For an orbit of period $2$ we need
to consider the function of $T_{r}\circ T_{r}(x)$ - abbreviated $%
T_{r}^{2}(x) $ - and the associated dynamic equation

\begin{eqnarray}
T_{r}^{2}(x) &=&T_{r}\circ T_{r}(x)  \label{7} \\
&=&r^{2}x(1-x)(1-rx(1-x)).  \nonumber
\end{eqnarray}
This is again a nonlinear system and its dynamic behavior can be studied as $%
r$ varies using the same analysis as before. In particular, the fixed points
of $T_{r}^{2}(x)$ can be found by equating $T_{r}^{2}(x)$ with $x$ and
solving the resulting $4$th order equation. Hence

\begin{eqnarray*}
x &=&T_{r}^{2}(x) \\
&=&r^{2}x(1-x)(1-rx(1-x)) \\
&=&-r^{3}x^{4}+2r^{3}x^{3}-(r^{2}+r^{3})x^{2}+r^{2}x
\end{eqnarray*}
whence we can derive the four fixed points, namely:

\begin{eqnarray*}
\overline{x}_{1} &=&0 \\
\overline{x}_{2} &=&1-1/r \\
\overline{x}_{3} &=&\frac{1}{2r}\left( r+1+\sqrt{(r-3)(r+1)}\right) \\
\overline{x}_{4} &=&\frac{1}{2r}\left( r+1-\sqrt{(r-3)(r+1)}\right)
\end{eqnarray*}
Clearly, the four fixed points of $T_{r}^{2}(x)$ are the two fixed points of 
$T_{r}(x)$ and the two elements of the $2$-cycle, which have no counterpart
in $T_{r}(x)$ - see the phase diagram in Figure $5$(a).

The fixed points of the second-order system ($7$) are characterized by the
derivative of $T_{r}^{2}(x),$ $(T_{r}^{2})^{\prime }(x).$ Since $%
(T_{r}^{2})^{\prime }(0)=r^{2}$ and $(T_{r}^{2})^{\prime }(1-1/r)=(2-r)^{2},$
for values of $r$ between $3$ and $3.45$, each of the fixed points $%
\overline{x}=0$ and $\overline{x}=1-1/r$ (which are still present) are
unstable. The other two fixed points, however, $\overline{x}=\frac{1}{2r}%
\left( r+1\pm \sqrt{(r-3)(r+1)}\right) $, are both stable, thus implying
that each of them locally attracts the dynamics of the second-order system ($%
7$).

With respect to Figure $5$(a), for $r$ between $3$ and $3.45$, the
trajectories of the first-order system ($3$) no longer converge to the fixed
point $\overline{x}=1-1/r$ (point B), but escape from it and diverge towards
the pair of fixed points, $\overline{x}=\frac{1}{2r}\left( r+1\pm \sqrt{%
(r-3)(r+1)}\right) $ - points D and C, respectively. Any one of them is
unstable under the first-order system ($3$) - since $|T_{r}^{\prime }(x)|>1$
at both C and D, so that the trajectories once in any one of these points
are initially repelled. Points C and D, however, are stable under the
second-order system ($7$) - since $(T_{r}^{2})^{\prime }(x)$ at both C and D
is less than $1$ in absolute value, so that after having moved away from
each of C and D in the first step, trajectories come back to each of these
points in the second step, thus making the dynamics of system ($7$) stable
with respect to each of C and D. Summarizing, for $r$ between $3$ and $3.45$%
, the trajectories of $T_{r}(x)$ oscillate in the set $\{$C, D$\}$, giving
rise to a stable $2$-cycle for $T_{r}(x)$, as it is shown in Figure $5$(b).
In this case the system is said to undergo a {\it flip bifurcation} - see
John Guckenheimer and Philip Holmes (1983).

If $r$ is increased further, then the two stable fixed points of $%
T_{r}^{2}(x)$ become unstable. In particular, both fixed points of $%
T_{r}^{2}(x)$ will bifurcate at the same $r$ value, leading to an orbit of
period $4$. In other words

\begin{eqnarray*}
T_{r}^{2}\circ T_{r}^{2}(x) &=&T_{r}^{4}(x) \\
&=&T_{r}\circ T_{r}\circ T_{r}\circ T_{r}(x)
\end{eqnarray*}
will have eight fixed points, four of which will be stable. For example, for 
$r=3.498561699$ there is a superstable orbit of period $4:0.5$, $0.874...$, $%
0.383...$, and $0.827...$ - see the phase and state space representations in
Figures $6$(a) and $6$(b).

The same bifurcation scenario will repeat over and over again as $r$ is
increased, yielding orbits of period $16$, $32$, $64,$ and so on {\it ad
infinitum}. However, the sequence $\{r_{\kappa }\}$ of values of $r$ at
which $\kappa $-cycles appear has a finite accumulation point $r_{\infty
}\simeq 3.569946$, involving an infinite number of period doubling
bifurcations\footnote{%
The values of $r$ for which these transitions from one cycle to another
cycle occur, are called {\it bifurcation points}, the transitions are called 
{\it bifurcations}, and the phenomenon is called {\it period-doubling}.}.
The limit set corresponding to $r_{\infty }$ is a geometric object with a
non-integer fractal dimension $\simeq 0.538$ and a Lyapunov exponent equal
to zero, and consequently the motion on it is not chaotic in the sense
defined above. In fact Mitchell Feigenbaum (1978) discovered that
convergence of $r$ to $r_{\infty }$ is controlled by the universal parameter 
$\delta \simeq 4.669202$ - known as the {\it Feigenbaum attractor}. The
computation of $\delta $ is based on the formula

\[
\delta =\lim_{\kappa \rightarrow \infty }\left( \frac{r_{\kappa }-r_{\kappa
-1}}{r_{\kappa +1}-r_{\kappa }}\right) 
\]
where $(r_{\kappa }-r_{\kappa -1})$ and ($r_{\kappa +1}-r_{\kappa })$ are
the distances on the real line between successive flip bifurcations.

Past $r_{\infty },$ we enter what is usually called the `chaotic zone'. For $%
r_{\infty }<r<4$ the model will behave either periodically or aperiodically
- in the latter case, the dynamics may be nonchaotic (zero Lyapunov
exponent, no sensitive dependence on initial conditions) or chaotic
(positive Lyapunov exponent, sensitive dependence on initial conditions).
There is, for example, a tiny interval near $r=3.83$ (a so-called {\it %
window of stability} or {\it periodicity}) where a stable $3$-cycle occurs -
see Figures $7$(a) and $7$(b). Just past $r=3.83$, the period doubling
occurs again, leading to orbits of period $6,$ $12,$ $24$, and so on, also
governed by the Feigenbaum constant. In fact, for $r$ between $r_{\infty 
\text{ }}$and $4$ there is a denumerably infinite number of periodic windows
and still an indenumerable number of values of $r$ for which the model
behaves aperiodically (chaotically or not). For $r=4$, we have a completely
chaotic orbit, as is illustrated in the state space diagram of Figure $8$.

In fact, the different period lengths $\kappa $ of stable periodic orbits
appear in a universal order, with higher-period cycles being associated with
higher values of $r$. In particular, if $r_{\kappa }$ is the value of $r$ at
which a stable $\kappa $-cycle first appears as $r$ is increased, then $%
r_{\kappa }>r_{q}$ if $\kappa \succ q$ (where $\kappa \succ q$ simply means
that ``$\kappa $ is listed before $q$'') in the following Sharkovski (1964)
ordering (in which we first list the odd numbers except one, then $2$ times
the odds, $2^{2}$ times the odds, etc., and at the end the powers of $2$ in
decreasing order - representing the period doubling)

\begin{eqnarray*}
3 &\succ &5\succ ...\succ 2\cdot 3\succ 2\cdot 5\succ ...\succ 2^{2}\cdot
3\succ 2^{2}\cdot 5\succ ... \\
&\succ &2^{3}\cdot 3\succ 2^{3}\cdot 5\succ ...\succ 2^{3}\succ 2^{2}\succ
2\succ 1
\end{eqnarray*}
This ordering seems strange, but it turns out to be the ordering which
expresses which periods imply which other periods. For example, the minimum $%
r$ value for an orbit of period $\kappa =2\cdot 3=6$ is larger than the
minimum $r$ value for an orbit of period $\kappa =2^{2}\cdot 3=12$, because $%
6\succ 12$ in the Sharkovski ordering. One consequence of this ordering is
that the existence of a stable $\kappa $ $(=3)$-cycle guarantees the
existence of any other stable $q$-cycle for some $r_{q}<r_{\kappa }$ - see,
for example, Tien-Yien Li and James Yorke (1975).

\section{The Ergodic Approach}

We have so far been discussing dynamical systems mainly from a geometric (or
topological) point of view. This approach, being intuitively appealing and
lending itself to suggestive graphical representations, has been
tremendously successful in the study of low-dimensional systems, such as,
for example, (discrete- and continuous-time) systems with one and perhaps
two variables. For higher-dimensional systems, however, the geometric
approach has encountered rather formidable obstacles and rigorous results
and classifications are few.

Thus, it is sometimes convenient to change perspective and adopt a different
approach, based on the axiomatic formulation of probability theory and aimed
at the investigation of statistical properties of orbits. This requires the
use and understanding of some basic notions and results of set theory and
measure theory, which we shall briefly review. We shall see that in many
aspects the ergodic theory of dynamical systems parallels the geometric one.
Moreover, the ergodic approach is more powerful and effective in dealing
with basic questions such as complexity and predictability as well as with
the relation between deterministic and stochastic systems.

\subsection{Some Elementary Measure Theory}

\begin{definition}
Let $X$ be a set of points $x$. A system $\Im $ of subsets of $X$ is called
a $\sigma $-algebra if

\begin{enumerate}
\item[a.]  $\emptyset ,X\in \Im $;

\item[b.]  $A\in \Im $ implies $A^{c}\in \Im $; and

\item[c.]  $A_{n}\in \Im ,n=1,2,...$, implies $\cup A_{n}\in \Im ,\cap
A_{n}\in \Im $.
\end{enumerate}
\end{definition}

\noindent That is, $\Im $ is a $\sigma $-algebra if the null set ($\emptyset 
$) and $X$ are in $\Im $, the set $A$ and its compliment ($A^{c}$) are in $%
\Im $, and given a sequence $\left\{ A_{n}\right\} _{n=1}^{\infty }$ of
subsets of $X$, $A_{n}\in \Im $, then the union $\cup A_{n}$ and the
intersection $\cap A_{n}$ are in $\Im $. The space $X$ together with a $%
\sigma $-algebra $\Im $ of its subsets is a {\it measurable space}, and is
denoted by $(X,\Im )$. Since we are dealing here with metric spaces (i.e.,
with spaces endowed with a distance, such as ${\Bbb R}^{n}$), among the
various $\sigma $-algebras, we shall consider the {\it Borel}-$\sigma $-{\it %
algebra,} i.e., the smallest such algebra containing the collections of open
(or closed) subsets of $X$.

\begin{definition}
Let $(X,\Im )$ be a measurable space. A real-valued function $\mu =\mu
(A),A\in \Im $, taking values in $[0,\infty ]$, is a measure if

\begin{enumerate}
\item[a.]  $\mu (\emptyset )=0$;

\item[b.]  $\mu (A)\geq 0$ for all $A\in \Im $; and

\item[c.]  if $\left\{ A_{n}\right\} _{n=1}^{\infty }$ is a disjoint
sequence of $\Im $-sets, then $\mu (\cup _{n=1}^{\infty
}A_{n})=\sum_{n=1}^{\infty }\mu (A_{n}).$
\end{enumerate}
\end{definition}

\noindent Thus a measure assigns zero to the empty set, is nonnegative, and
is countably additive. The triple $(X,\Im ,\mu )$ is called a {\it measure
space}. We shall be interested in finite measures (that is, $0\leq \mu
<\infty $) in which case $\mu $ can suitably be rescaled so that $\mu (X)=1$%
. When $\mu (X)=1$, $\mu (X)$ is called a {\it probability measure} and ($%
X,\Im ,\mu $) is called a {\it probability space}. In this case $X$ is the
sample space (or space) of elementary events, the sets $A$ in $\Im $ are
events, and $\mu (A)$ is the probability of the event $A.$

Two especially important examples of probability measures, which will be
used in the sequel, are the {\it Dirac} and the {\it Lebesque measures}. The
former, also called {\it Dirac delta} and usually denoted by $\delta _{x}$
is the probability measure that assigns value 1 to all the subsets $A$ of $X$
that contain a given point $x$, and value zero to all the subsets that do
not contain it. Formally, $\delta _{x}(A)=1$ if $x\in A$ and $\delta
_{x}(A)=0$ if $x\notin A.$ The {\it Lebesque measure} on the real line
(henceforth denoted by $m$) is the measure that assigns to each interval its
length as its measure\footnote{%
Thus, the Lebesgue measure corresponds to the intuitive notion of length
(for one-dimensional sets) and volume (for $k$-dimensional ones). It also
provides an intuitive and physically relevant notion of probability.}. In
particular, the Lebesque measure of $(a,b)$, as well as of any of the
intervals $(a,b]$, $[a,b]$, or $[a,b)$, is simply its length $\left|
b-a\right| $.

\begin{definition}
Let ($X,\Im ,\mu $) be a probability space. A {\it transformation} $T$ of $X$
into $X$ is measurable if, for every $A\in \Im $, $T^{-1}A=\left\{ x:Tx\in
A\right\} \in \Im $.
\end{definition}

\noindent Notice that $T^{-1}A$ denotes the pre-image of $A$ -- the set of
points that are mapped onto $A$ by $T$ in one step.

\begin{definition}
A measurable transformation $T$ is said to preserve a measure $\mu $ if for
every $A\in \Im $, $\mu (T^{-1}A)=\mu (A)$. If $T$ is measure-preserving
(with respect to $\mu $), $\mu $ is called $T$-invariant.
\end{definition}

When these concepts are applied to the investigation of dynamical systems, $%
X $ will typically correspond to the phase space and the elements $x$ of $X$
to states (or positions) of the system. The subsets $A$ (the events) will
correspond to certain interesting configurations of orbits of the system in
the phase space (such as, for example, fixed points, limit cycles, or
strange attractors, basins of attraction, etc.). Finally, the transformation 
$T$ will correspond to the (flow) map governing the evolution of the state
of the system in time\footnote{%
We can think, for example, of the transformation $T$ as a mechanism for
recursively generating a sample $\{x,Tx,...,T^{n}x\}$ from the domain $X$
from an initial condition $x\in X$. Since, in the present context, there is
no essential difference between discrete- and continuous-time dynamical
systems, in what follows we shall discuss the issue in terms of maps, i.e.,
discrete-time systems.}. We shall refer to a measure-preserving
transformation $T$ on the probability space ($X,\Im ,\mu $) as a {\it %
dynamical system}, denoted by ($X,\Im ,\mu ,T$).

\subsection{Ergodicity}

As we want to study the statistical properties of orbits generated by
measure-preserving transformations, we need to calculate averages over time.
In fact, certain basic quantities such as Lyapunov exponents (which, as we
have seen above, measure the rate of divergence of nearby orbits) or metric
entropy (which, as we shall see below, measures the rate of information
production of observations of a system) can be looked at as time averages.

The existence of such averages is guaranteed by the following theorem.

\begin{theorem}
(Birkhoff and Khinchin).{\bf \ }Let{\bf \ }$(X,\Im ,\mu )$ be a probability
space, $T$ measure preserving and ergodic on $X$, and $f$ an integrable
function. Then 
\[
\lim_{n\rightarrow \infty }\frac{1}{n}\sum_{i=0}^{n-1}f(T^{i}x)=\widehat{f}%
(x) 
\]
exists for $\mu $-almost every $x\in X$\footnote{%
`$\mu $-almost every $x\in X$' means `all points $x\in X$, except a set of
points to which $\mu $ assigns zero value'.}. $\widehat{f}(x)$ is $T$%
-invariant, i.e., $\widehat{f}(T(x))=\widehat{f}(x).$
\end{theorem}

In general, time averages depend on $x$, meaning that they may be different
for orbits originating from different initial states. This happens, for
example, when the space $X$ is decomposable (under the action of $T$), in
the sense that there exist, say, two subspaces $X_{1}$ and $X_{2}$, both
invariant with respect to $T$ (i.e., $T$ maps points of $X_{1}$ only to $%
X_{1}$ and points of $X_{2}$ only to $X_{2})\footnote{%
The dynamic decomposability of the system - a geometric, or topological fact
- is reflected in the existence of $T$-invariant measures $\mu $ that are
decomposable in the sense that they can be represented as a weighted average
of invariant measures. For example, in the case mentioned above, we can
write $\mu =\alpha \mu _{1}+(1-\alpha )\mu _{2}$, where $\alpha \in (0,1)$
and $\mu _{1}$ and $\mu _{2}$ may or may not be further decomposed.}$. It is
for this reason that we shall concentrate in a fundamental class of
invariant measures that satisfy the requirement of indecomposability, and
known as {\it ergodic measures}. This will ensure that $X$ is (dynamically)
indecomposable - a requirement, for it to be called chaotic.

\begin{definition}
Given a dynamical system $(X,\Im ,\mu ,T)$, the measure-preserving
transformation $T$ is said to be ergodic (or indecomposable) if $%
T^{-1}(A)=A, $ for some $A\in \Im ,$ implies either $\mu (A)=1$ or $\mu
(A)=0.$ In this case, the $T$-invariant measure $\mu $ is also said to be an
ergodic measure for $T$.
\end{definition}

\noindent As an example, consider a discrete-time dynamical system
characterized by an attracting periodic orbit of period $k,\left\{
x,Tx,...,T^kx=x\right\} $. In this case, the measure that assigns the value $%
1/k$ to each point of the orbit is invariant and ergodic.

To discuss ergodic properties of dynamical systems, let $f$:$X\rightarrow 
{\Bbb R}$ be a measurable function, representing a measurement made on the
system (such as, for example, the number of times that an orbit generated by
the map $T$ and starting from the point $x$ visits a set $A$ when $T$ is
iterated $n$ times). As it was argued in the previous section, it is
interesting and sometimes necessary to consider the {\it time average} of $f$
(the average value of $f$ evaluated along the forward trajectory), defined by

\[
\widehat{f}(x)=\lim_{n\rightarrow \infty }\frac{1}{n}%
\sum_{i=0}^{n-1}f(T^{i}x). 
\]
If $\widehat{f}(x)$ exists, it may be thought of as an equilibrium value of $%
f$.

Alternatively, we could evaluate the {\it space} (or {\it phase}) {\it %
average} of $f$, by considering $f$ as a function of $x$ and multiplying
that value by the probability that the system visits the set $A$. This
average is the expectation (or mean value) of $f(x)$ evaluated on the space $%
X$

\[
\overline{f}=\int_{X}f(x)d\mu (x). 
\]
The following result establishes the connection between the time average and
the space average.

\begin{theorem}
Let{\bf \ }$(X,\Im ,\mu )$ be a probability space. If the measure-preserving
transformation $T$ is ergodic then the limit function $\widehat{f}(x)$
defined in the Birkhoff-Khinchin theorem above is a constant and we have 
\[
\lim_{n\rightarrow \infty }\frac{1}{n}\sum_{i=0}^{n-1}f(T^{i}x)=\widehat{f}%
(x)=\int_{X}f(x)d\mu (x) 
\]
\end{theorem}

\noindent In words, the theorem states that if $T$ is ergodic, then the time
average, $\widehat{f}$ equals the space average $\overline{f}$.

\subsection{Lyapunov Exponents Revisited}

Clearly, ergodic theory provides an alternative (and often simpler) way of
calculating average properties of a dynamical system. In fact, we can use
the ideas just discussed, to reformulate the definition of Lyapunov
exponents. In particular, if we choose $f(x)=\log \left| T^{\prime
}(x)\right| $ and apply the ergodic theorem, then the Lyapunov exponent of a
map $T$ can be written as 
\[
\lambda =\int_{X}\text{log}\left| T^{\prime }(x)\right| \text{d}\mu (x) 
\]
which is independent of the initial condition. The quantity log$\left|
T^{\prime }(x)\right| $ - the natural logarithm of the absolute value of the
slope of the curve generated by the map $T$ in the $(x_{t+1},x_{t})$ plane -
measures the (exponential) rate at which small discrepancies between
trajectories (or small errors) are amplified by the action of the map. In
the general case in which that slope varies with $x$, its different values
are weighted by $\mu $. It follows that slopes obtaining over sets of values
of $x$ whose $\mu $-measure is zero, do not affect the final results.

All this can be easily illustrated by two simple examples. Let us first
compute the Lyapunov exponent for the asymmetric tent map 
\[
T_{\hat{\Lambda}}(x)=\left\{ 
\begin{array}{cc}
x/a & \text{{\rm for} }0\leq x\leq a \\ 
(1-x)/(1-a) & \text{{\rm for} }a\leq x\leq 1
\end{array}
\right. 
\]
In this case, it is easy to see that the map $T_{\hat{\Lambda}}(x)$
preserves the Lebesque measure whose density function is $\rho (x)=1$,
implying that $\mu (x)=\int_{x}\rho (x)dx=\int_{x}dx$ and $d\mu (x)=dx$.
Consequently, 
\begin{eqnarray*}
\lambda &=&\int_{0}^{1}\log \left| T_{\hat{\Lambda}}^{\prime }(x)\right|
d\mu (x) \\
&=&\int_{0}^{a}\log \left( \frac{1}{a}\right) dx+\int_{a}^{1}\log \left( 
\frac{1}{1-a}\right) dx \\
&=&a\log (\frac{1}{a})+(1-a)\log (\frac{1}{1-a})
\end{eqnarray*}
Clearly, for $a=1/2$, we are in the case of the symmetric tent map, $%
T_{\Lambda }(x)$, and $\lambda =\log 2.$

As another example, let us compute the Lyapunov exponent for the logistic
map (for $r=4$), $T_{4}(x)=4x(1-x)$, using its conjugacy with the symmetric
tent map, $T_{\Lambda }(x)$, $h(x)=\sin ^{2}(\pi x/2)$. Since the tent map
preserves Lebesque measure, the conjugacy also induces an invariant measure $%
\mu (x)$ for the logistic map, with density function $\rho (x)=\pi
^{-1}[x(1-x)]^{-1/2}$. Consequently, the Lyapunov exponent for the logistic
map (for $r=4$) is given by\footnote{%
Recall that we are using the invariant measure $\mu (x)=\int_{x}\rho
(x)dx=\int_{x}\frac{1}{\pi [x(1-x)]^{1/2}}dx$, which implies that $d\mu (x)=%
\frac{1}{\pi [x(1-x)]^{1/2}}dx.$} 
\begin{eqnarray*}
\lambda &=&\int_{0}^{1}\log \left| T_{4}^{\prime }(x)\right| \frac{\text{d}x%
}{\pi [x(1-x)]^{1/2}} \\
&=&\int_{0}^{1}\frac{\log \left| 4-8x\right| }{\pi [x(1-x)]^{1/2}}dx \\
&=&\log 2.
\end{eqnarray*}

\section{Predictability, Entropy}

The rather formidable apparatus described above will allow us to discuss the
question of predictability in a rigorous manner. In so doing, however,we
must first remove a possible source of confusion. In particular, given that
the ergodic approach analyzes dynamical systems by means of probabilistic
methods, one might immediately point out that since the outcomes of
deterministic dynamical systems are not random events, measure and
probability theories are not the appropriate tools of analysis. {\it Prima
facie, } this seems to be a convincing argument -- if the system is
deterministic, we know the equations of motion, and we can measure its state
with infinite precision, then there is nothing left to discuss.

However, infinite precision of observation is a purely mathematical
expression, and it has no physical counterpart. When dynamical system theory
is applied to real systems, a distinction must be made between {\it states}
of a system, i.e., points in a state space, and {\it observable states},
i.e., subsets (or cells) of the state space, whose (non-zero) size reflects
our limited power of observation\footnote{%
As will become apparent in the discussion that follows, this distinction is
not important for systems whose orbit structure is simple, such as, for
example, systems characterized by a stable fixed point or a stable limit
cycle. For these systems, that is, the (unrealistic) assumption of infinite
precision of observation is a convenient simplification.}. This will be
consistent with the fact that in real systems perfect foresight only makes
sense when it is interpreted as an asymptotic state of affairs which is
approached as economic agents accumulate information and learn about the
position of the system. Much of what follows concerns the conditions under
which, given precise knowledge of the equations of the system (i.e., given a
deterministic system), but an imprecise, albeit arbitrarily accurate,
observation of its state, prediction is possible.

We can now characterize the concept of predictability in a rigorous fashion,
in terms of a quantity called entropy. Let $(p_{1},...,p_{N})$ be a finite
probability distribution, i.e., $p_{i}\geq 0$ for all $i$ and $%
p_{1}+...+p_{N}=1$, for the occurrence of events $A_{1},...,A_{N}$. The
entropy of this distribution is

\[
H=-\sum_{i=1}^{N}p_{i}\log (p_{i}), 
\]
with $0\hbox{log}0=0$. $H$ measures the degree of indeterminacy
(uncertainty) of an event. It attains its largest value $(\hbox{log}N)$ for $%
p_{1}=...=p_{N}=1/N$, meaning that the distribution has maximal
indeterminacy, and its minimum value (zero) when one of the $p$'s is equal
to one, the others being zero\footnote{%
For example, in a game of dice, the maximum entropy of a throw (the maximum
uncertainty about its outcome) obtains when each of the six facets of a die
has the same probability $(1/6)$ of turning up. An unfair player can reduce
the uncertainty of the outcome by `loading' the dice and thereby increasing
the probability of one or more of the six faces (and correspondingly
decreasing the probability of the others).}.

We shall now apply this entropy idea to a description of the state space
behavior of a dynamical system, $(X,\Im ,\mu ,T)$. Let a single trajectory
run for a long time to map out an attractor, and let ${\cal P}=(P_{1},\ldots
,P_{N})$ be a finite $\mu $-measurable partition of $X\footnote{%
A partition can also be viewed as a function ${\cal P}:X\rightarrow \left\{
P_{1},...,P_{N}\right\} $ such that, for each point of the state space $x\in
X$, ${\cal P}(x)$ is the element of the partition, the cell of $X$, in which 
$x$ is contained.}$. The entropy of ${\cal P}$ will be equal to

\[
H({\cal P})=-\sum_{i=1}^{N}\mu (P_{i})\log \mu (P_{i}), 
\]
where $\mu (P_{i})$ measures the probability of finding the system in the
`cell' $P_{i}$.

However, when dealing with a dynamical system, we are not interested in the
entropy of a partition of the state space (i.e., the information in a single
experiment), but in the entropy of the system (i.e., the rate at which
replications of the experiment produce information). To make this idea more
precise, for each $P_{i}$, we write $T^{-k}P_{i}$ for the set of points that
led to $P_{i}$ in $k$ steps. We then denote by $T^{-k}{\cal P}$ the
partition $(T^{-k}P_{1},...,T^{-k}P_{N})$, which is deduced from ${\cal P}$
by time evolution. Finally, we define the `super-partition'\footnote{%
Given two partitions ${\cal P}_{1\text{ }}$and ${\cal P}_{2}$, the operation 
${\cal P}_{1}\bigvee {\cal P}_{2}$ consists of all the possible
intersections of the elements of ${\cal P}_{1\text{ }}$and ${\cal P}_{2}$
and it is called a `span'.}

\[
\bigvee\limits_{i=1}^{n}T^{-i}{\cal P}\equiv {\cal P}\vee T^{-1}{\cal P}\vee
\cdots \vee T^{-n}{\cal P} 
\]
which is generated by ${\cal P}$ in a time interval of length $n$. The
entropy of the super-partition, $\bigvee_{i=1}^{n}T^{-i}{\cal P}$, namely $%
H(\bigvee_{i=1}^{n}T^{-i}{\cal P)}$, can be calculated analogously, summing
over all the cells of $\bigvee_{i=1}^{n}T^{-i}{\cal P}$. A moment's
reflection will suggest that, whereas an element of the original partition $%
{\cal P}$ corresponds to a (approximately observed) state of a dynamical
system, an element of the super-partition, $\bigvee_{i=1}^{n}T^{-i}{\cal P}$%
, corresponds to a {\it sequence} of $n$ states.

If we now divide $H(\bigvee_{i=1}^{n}T^{-i}{\cal P)}$ by the number of
observations $n$, we obtain the average amount of information contained in -
the average amount of uncertainty about - the `super-experiment' consisting
in the repeated observation of the system along a typical orbit. If we
increase the number of observations indefinitely, we obtain\footnote{%
See Patrick Billingsley (1965, pp. 81-82) or Ricardo Ma\~{n}\'{e} (1987, pp.
216) for a proof that this limit exists.} 
\[
h(T,{\cal P})=\lim_{n\to \infty }{\frac{1}{n}}H(\bigvee_{i=1}^{n}T^{-i}{\cal %
P}), 
\]
which is the entropy of the transformation $T$ with respect to the partition 
${\cal P}$\footnote{%
In the literature, we also find the expression $h(\mu ,{\cal P})$ where the
system is identified by the invariant measure. In the present context, the
two exressions are entirely equivalent.}.

From the definitions above, the link between entropy and predictability
should be clear. Zero entropy means that, if we observe the state of a
dynamical system long enough, although with finite precision (and we know
the ``true'' law of motion), then there is no uncertainty left about the
future. On the contrary, positive entropy means that, no matter how long we
observe the system, additional observations are informative, i.e., the
future is unpredictable.

To investigate this point a little further, $h(T,{\cal P})$ can be looked at
as the limit of a fraction - the numerator is the entropy of a
`super-partition' obtained by iterating $T$, and the denominator is the
number of iterations. Loosely speaking, if when the number of iterations
increases, $H(\bigvee_{i=1}^{n}T^{-i}{\cal P})$ remains bounded, the limit
will be zero; if it grows linearly with $n$ the limit will be a finite,
positive value; if it grows more than linearly, the limit will be unbounded.
To interpret this result, consider that each cell of the partition $%
\bigvee_{i=1}^{n}T^{-i}{\cal P}$ corresponds to a sequence of length $n$ of
cells of ${\cal P}$ (i.e., to an orbit of length $n$ of the system, observed
with ${\cal P}$-precision). Considering the definition of entropy, one can
verify that $H(\bigvee_{i=1}^{n}T^{-i}{\cal P})$ will increase with $n$
linearly according to whether, increasing the number of observations, the
number of possible sequences will increase {\it exponentially}. From this
point of view, it is easy to understand why `simple' systems (i.e., those
characterized by attractors which are fixed points or periodic orbits) have
zero entropy. Transients apart, for those systems the possible sequences of
states are limited and their number does not increases at all with the
number of observations. Complex systems are precisely those for which the
number of possible sequences of states grows exponentially with the number
of observations. For finite-dimensional, deterministic systems characterized
by bounded attractors, entropy is bounded above by the sum of the positive
Lyapunov exponents and is therefore finite.\footnote{%
The entropy of a system with respect to a given partition can be given an
alternative, very illuminating formulation by making use of the auxiliary
concept of {\it conditional entropy} of (partition) ${\cal A}$ given
(partition) ${\cal B}$, defined by
\par
\[
H({\cal A}|{\cal B})=-\sum_{A,B}\mu (A\cap B)\log \ \mu (A|B) 
\]
where $A,B$ denote elements of the partitions ${\cal A}$ and ${\cal B}$,
respectively. Intuitively, conditional entropy can be viewed as the average
amount of uncertainty of the experiment ${\cal A}$ when the outcome of the
experiment ${\cal B}$ is known. It can be shown [see Billingsley (1965, pp.
81-82)] that
\par
\[
\lim_{n\to \infty }{\frac{1}{n}}H(\bigvee_{i=1}^{n}T^{-i}{\cal P}%
)=\lim_{n\to \infty }H({\cal P}\left| \bigvee_{i=1}^{n}T^{-i}{\cal P}%
)\right. 
\]
This equation provides another useful interpretation of $h(T,{\cal P})$: it
is \QTR{textit}{the amount of uncertainty of - the amount of information
contained in - an observation of the system in the partitioned state space,
conditional upon the (finite-precision) knowledge of its state in the
infinitely remote past}.}.

So far we have been talking about entropy relative to a specific partition.
The entropy of a transformation $T$, or equivalently the entropy of the $T$%
-invariant measure $\mu $, is 
\[
h(T)=\hbox{Sup}_{{\cal P}}\ h(T,{\cal P}) 
\]
where the supremum is taken over all finite partitions. The quantity $h(T)$
is also known as {\it K}(olmogorov)-{\it S}(inai), or {\it metric entropy}.
Unless we indicate differently, by entropy we mean K-S entropy. Actual
computation of K-S entropy, $h(T)$, directly from its definition looks a
rather desperate project. Fortunately, a result from Kolmogorov and Sinai
guarantees that, under conditions often verified in specific problems, the
entropy of a system $h(T)$ can be obtained from the computation of its
entropy relative to a given partition, $h(T,{\cal P})$. Formally, we have
the following\footnote{%
For discussion and proof of the K-S theorem, see Billinglsey (1965, pp.
84-85) or Ma\~{n}\'{e} (1987, pp. 218-22).}

\begin{theorem}
(Kolmogorov-Sinai). \QTR{textit}{Let $(X,\Im ,\mu )$ be a probability space; 
$T$ a transformation preserving $\mu $ and ${\cal P}$ a partition of $(X,\Im
,\mu )$ with finite entropy. If $\bigvee_{i=1}^{\infty }T^{-i}{\cal P}=\Im \ 
\hbox{mod
0}$, then $h(T)=h(T,{\cal P})$. In this case, ${\cal P}$ is called a {\rm %
generator}.}
\end{theorem}

As an example, consider the symmetric tent map given by equation (?) and the
partition of the interval consisting of the two sub-intervals located,
respectively to the left and to the right of the $1/2$ point. Thus, we have
a partition ${\cal P}=\{P_{1},P_{2}\}$ of $[0,1]$, where $P_{1}=\{0<x<1/2\}$
and $P_{2}=\{1/2<x<1\}.$ Then the atoms of $T^{-1}P_{1}$ are the two
subintervals $\{0<x<1/4\}$ and $\{3/4<x<1\}$ and the atoms of $T^{-1}P_{2}$
are the two subintervals $\{1/4<x<1/2\}{\bf \ }$and $\{1/2<x<3/4\}.$ Hence,
taking all possible intersections of subintervals, the span $\{T^{-1}{\cal P}%
\bigvee {\cal P}\}$ consists of the four subintervals $\{0<x<1/4\}$, $%
\{1/4<x<1/2\}$, $\{1/2<x<3/4\}$, and $\{3/4<x<1\}$. Repeating the same
operation $m$ times the span $\{\bigvee_{i=0}^{m-1}T^{-i}{\cal P}$ $\}$ is
formed by $2^{m}$ subintervals of equal length $2^{-m}$, defined by $%
\{x:(i-1)/2^{m}<x<i/2^{m}\},\ 1\leq i\leq 2^{m}$. Moreover, considering that
(in the case of the tent map) the span $\bigvee_{i=0}^{\infty }T^{-i}{\cal P}
$ contains any open subinterval of $[0,1]$ and therefore, if we use the
Borel $\sigma $-algebra, the selected partition is a generator, we can apply
the Kolmogorov-Sinai Theorem and have $h(T)=h(T,{\cal P)}$. Finally, taking
into account the fact that the tent map preserves the Lebesgue measure $m$
(which, we recall, assigns to each measurable set a value equal to its
length), we conclude that the K-S entropy of the tent map is 
\begin{eqnarray*}
h(T) &=&\lim_{m\rightarrow \infty }{\frac{1}{m}}H\left(
\bigvee_{i=0}^{m-1}T^{-i}{\cal P}\right) \\
&=&\lim_{m\rightarrow \infty }\frac{1}{m}\left[ -2^{m}(2^{-m}\log
(2^{-m}))\right] \\
&=&\log 2.
\end{eqnarray*}

Before concluding this section, we would like to notice that entropy is
closely linked with another type of statistical invariants, the Lyapunov
exponents. It can be shown that in general we have the following inequality 
\[
h(T)\leq \sum_{i:\lambda _{i}>0}\lambda _{i}, 
\]
where $\lambda $ denotes a Lyapunov exponent. For\QTR{textit}{\ observable}
chaotic systems, strict equality holds\footnote{%
For technical details, see Donald Ornstein and Benjamin Weiss (1991, pp.
78-85) or Ruelle (1989, pp. 71-77).}. As we have seen before, the equality
indeed holds for the tent map.

The close relation between entropy and Lyapunov exponents is not surprising.
We have already observed that entropy crucially depends on the rate at which
the number of new possible sequences of `coarsed-grained' states of the
system grows as the number of observations increases. But this rate is
strictly related to the rate of divergence of nearby orbits, which is
measured by the Lyapunov exponents. Thus, the presence of one positive
Lyapunov exponent on the attractor signals positive entropy and
unpredictability of the system.

\section{Isomorphism}

In the discussion of dynamical systems from a geometric point of view, we
have encountered the notion of topological equivalence. Analogously, there
exists a fundamental type of equivalence relation between measure-preserving
transformations, called {\it isomorphism,}{\bf \ }which plays a very
important role in ergodic theory and which we shall use in the sequel.

\begin{definition}
Two transformations $T$ and $\widehat{T}$ acting, respectively, on the state
spaces $X$ and $\widehat{X}$, and preserving, respectively, the measures $%
\mu $ and $\widehat{\mu }$, are isomorphic if a one-to-one and invertible
map $\theta $ exists such that (excluding perhaps certain sets of measure
zero)

\begin{enumerate}
\item[a.]  $\widehat{T}\circ \theta =\theta \circ T$; and

\item[b.]  the map $\theta $ preserves the probability structure, i.e., if $%
I $ and $\widehat{I}$ are, respectively, measurable subsets of $X$ and $%
\widehat{X}$, then $\mu (I)=\widehat{\mu }\circ \theta (I)$ and $\widehat{%
\mu }(\widehat{I})=m\circ \theta ^{-1}(\widehat{I})$.
\end{enumerate}
\end{definition}

\noindent Certain properties such as ergodicity and entropy are invariant
under isomorphism. Consequently, isomorphic transformations have the same
entropy\footnote{%
The reverse is true only for a certain class of transformations called
Bernoulli.}.

As an example, we will show that the logistic map (with $r=4$), $T_{4}(x)$,
and the tent map, $T_{\Lambda }(x)$, are isomorphic, and therefore have the
same entropy. By making use of the topological conjugacy, $\theta (x)=\sin
^{2}(\pi x/2)$, we have already shown that $T_{4}(x)$ and $T_{\Lambda }(x)$
are topologically conjugate, that is $\theta \circ T_{4}(x)=T_{\Lambda
}(x)\circ \theta $. In general, however, topological conjugacy need not
imply measure-theoretic isomorphism. So we still have to prove that $%
T_{4}(x) $ preserves a measure $\rho $ such that, for almost all
subintervals $I$ of $[0,1]$, $\rho (I)=m(\theta (I))$, where $m$ is the
Lebesgue measure which, as we already know, is $T_{\Lambda }(x)$-invariant.

If these measures are absolutely continuous, the last equation is equivalent
to $\rho (dx)=(d\theta /dx)dx$, whence, making use of the definition of $%
\theta (x)$ and considering that $\hbox{cos}(\theta )=[1-\hbox{sin}%
^{2}(\theta )]^{1/2}$, we obtain 
\[
\rho (dx)={\frac{dx}{\pi [x(1-x)]^{1/2}}}. 
\]
Now, considering that the counter-image of each interval $I$ under $%
T_{\Lambda }(x)$ consists of two subintervals whose length is half the
length of $I$, it is easily verified that $T_{\Lambda }(x)$ preserves the
Lebesgue measure. If we also consider that $T_{4}^{-1}(x)=\theta ^{-1}\circ
T_{\Lambda }^{-1}(x)\circ \theta $, we can establish that $T_{4}(x)$
preserves $m\theta $ and therefore it preserves $\rho $. Since isomorphism
preserves entropy, we can conclude that the logistic map $T_{4}(x)$, has
entropy equal to $\hbox{log}2$ $>$ $0$ and its outcome is therefore
unpredictable\footnote{%
Notice that, in this case, the metric entropy and the unique Lyapunov
exponent are equal.}.

The implications for economics of the results just obtained are puzzling.
For example, consider the case in which models of optimal growth give rise
to dynamic, logistic-type equations with chaotic parameter. The sequences
thus generated are optimal in the sense that they solve a problem of
intertemporal maximization of rational agents, in an economy satisfying the
requirements of competitive equilibrium at each point of time. In the
absence of (exogenous) random disturbances, along optimal trajectories
agents' expectations are supposed to be always fulfilled. While the latter
assumption may be acceptable when the dynamics of the system are simple
(i.e., convergence to a steady state or to a periodic orbit), it makes
little sense if the dynamics are chaotic.

\section{Chaos in Dynamic Economic Models}

Chaos represents a radical change of perspective on business cycles.
Business cycles receive an endogenous explanation and are traced back to the
strong nonlinear deterministic structure that can pervade the economic
system. This is different from the (currently dominant) exogenous approach
to economic fluctuations, based on the assumption that economic equilibria
are determinate and intrinsically stable, so that in the absence of
continuing exogenous shocks the economy tends towards a steady state, but
because of stochastic shocks a stationary pattern of fluctuations is
observed.

Richard Goodwin was one of the first (back in the 1950's and 1960's) to
understand the relevance of chaos theory for economics - see Goodwin (1982),
for a collection of the relevant papers. Recently, however, there has been a
revival of interest in dynamical systems theory, and there is a group of
economists who look at economic fluctuations as deterministic phenomena,
endogenously created by market forces, and aggregator (utility and
production) functions. They agree with Goodwin that chaos theory has great
implications for both theory and policy. For example, chaos could help unify
different approaches to structural macroeconomics. As Jean-Michel Grandmont
(1985) has shown for different parameter values even the most classical of
economic models can produce stable solutions (characterizing classical
economics) or more complex solutions, such as cycles or even chaos
(characterizing much of Keynesian economics).

In what follows, we shall briefly review some representative theoretical
microeconomic and macroeconomic models that predict cycles and chaos as
outcomes of reasonable economic hypotheses. Our purpose is not to provide a
complete survey of all existing dynamic economic models that predict chaos.
The reader that is interested in a more exhaustive survey should also
consult William Brock (1988), Michele Boldrin and Michael Woodford (1992),
Kazuo Nishimura and Gerhard Sorger (1996), and Pietro Reichlin (1997).

\subsection{Rational Choice and Chaos}

Benhabib and Richard Day (1981), using a standard micro-framework, showed
that rational choice can lead to erratic behavior when preferences depend on
past experience. Following Benhabib and Day (1981), consider the
(logarithmic representation of the) Cobb-Douglas utility function

\[
u(x,y;\alpha )=\alpha \log x+(1-\alpha )\log y 
\]
with $0<\alpha <1.$ Maximizing subject to (the usual budget constraint)

\begin{equation}
p_{x}x+p_{y}y=I  \label{Budget Constraint}
\end{equation}
yields the Marshallian demand functions

\begin{equation}
x=\alpha \frac{I}{p_{x}}\text{\qquad and\qquad }y=(1-\alpha )\frac{I}{p_{y}}
\label{Marshallian}
\end{equation}

Assuming, however, that preferences depend on past experience, as in
Benhabib and Day (1981), according to a function

\begin{equation}
\alpha _{t}=rx_{t-1}y_{t-1}  \label{BD}
\end{equation}
where $r$ is an `experience dependence' parameter, then the demand for $x$
and $y$ is described by a first-order difference equation in $x$ and $y$,
respectively. For example, by substituting (\ref{BD}) into (\ref{Marshallian}%
) and exploiting the budget constraint (\ref{Budget Constraint}), the demand
for $x$ is obtained (under the assumption of constant prices) as

\begin{equation}
x_{t}=\frac{rI}{p_{x}p_{y}}x_{t-1}(I-p_{x}x_{t-1})  \label{BD Demands}
\end{equation}
Clearly, Equation (\ref{BD Demands}) describes a one-humped curve like the
logistic map (4). In fact, for $p_{x}=p_{y}=I=1$, Equation (\ref{BD Demands}%
) reduces to Equation (4). Therefore, the specification of experience
dependent preferences generates chaotic behavior for appropriate values of
the experience dependence parameter, $r.$

\subsection{Descriptive Growth Theory and Chaos}

Following Day (1982), we consider the descriptive one-sector model due to
Robert Solow\ (1956). Under the assumptions that aggregate saving equals
gross investment and that the capital stock exists for exactly one period,
this system can be written as a first-order system in discrete time as

\begin{equation}
(1+\nu )k_{t+1}=sf(k_{t})  \label{Descriptive growth}
\end{equation}
where $k$ is capital per worker, $f$ a neoclassical production function, and
the two parameters $\nu >-1$ and $s\in [0,1]$ represent, respectively, the
rates of population growth and saving. Under the usual convexity
assumptions, the phaseline of Equation (\ref{Descriptive growth}) is an
increasing concave function through the origin, with two fixed points. The
trivial steady state at $0$ is asymptotically unstable while the other
(positive) fixed point is globally stable, attracting orbits that start at
any initial value $k_{0}>0.$

Day (1982) extended the above neoclassical one-sector model of capital
accumulation by introducing a pollution effect that reduces productivity as
in the following (Cobb-Douglas type) production function

\begin{equation}
f(k_{t})=Bk_{t}^{\varphi }(\varsigma -k_{t})^{\gamma }  \label{Day}
\end{equation}
where $k_{t}\leq \varsigma =$ constant (acting as a saturation level of
capital per worker) and $(\varsigma -k_{t})^{\gamma }$ reflects the effect
of pollution on per capita output. In particular, when $k$ increases,
pollution also increases and less output can be produced with a given stock
of capital than in the standard model. With (\ref{Day}), the neoclassical
model (\ref{Descriptive growth}) becomes

\begin{equation}
(1+\nu )k_{t+1}=sBk_{t}^{\varphi }(\varsigma -k_{t})^{\gamma }
\label{Descriptive growth2}
\end{equation}
which for $B=\gamma =\varsigma =1$ reduces to

\begin{equation}
k_{t+1}=rk_{t}(1-k_{t})  \label{Descriptive growth3}
\end{equation}
where $r=sB/(1+\nu )$. Equation (\ref{Descriptive growth3}) is formally
identical with the logistic map (4). Hence, all properties of the logistic
map apply here as well. Moreover, the general five-parameter map (\ref
{Descriptive growth2}) is also chaotic for appropriate values of the
parameters - see Day (1982) or Lorenz (1993) for details.

\subsection{Optimal Growth Theory with Money and Chaos}

In this section we consider one version of the neoclassical growth model -
Miquel Sidrauski's (1967) optimal growth model with money. It is assumed
that the economy is composed of a large number of identical infinitely lived
households, each maximizing (at time $t$) a lifetime utility function of the
form

\[
\sum_{t=0}^{\infty }\beta ^{t}u(c_{t},m_{t}) 
\]
where $c$ and $m$ are consumption and real money balances per capita.
Ignoring capital accumulation, production, and interest-bearing public debt,
the representative household's budget constraint for period $t$ is assumed
to be

\[
P_{t}(m_{t}+c_{t})=P_{t}y+H_{t}+P_{t-1}m_{t-1} 
\]
where $y$ is a constant endowment and $H_{t}$ is per capita lump-sum
government transfers, assumed to be equal to $\mu M_{t-1}$ (where $\mu >0$
is the constant rate of money growth). Assuming additive instantaneous
utility, the equilibrium fixed points for the system are obtained by solving
the following first-order difference equation [see Costas Azariadis (1993,
section 26.3) for more details]

\begin{equation}
m_{t+1}u_{c}(y,m_{t+1})=\frac{1+\mu }{\beta }\left[
u_{c}(y,m_{t})-u_{m}(y,m_{t})\right] m_{t}  \label{OG1}
\end{equation}

If we drop the separability assumption and instead consider

\[
u(c,m)=\frac{\left( c^{1/2}m^{1/2}\right) ^{1-\sigma }}{1-\sigma },\qquad
\sigma >0,\sigma \neq 1 
\]
where $\sigma $ is the reciprocal of the intertemporal elasticity of
substitution between current and future values of the aggregate commodity $%
(cm)^{1/2}$, then Equation (\ref{OG1}) simplifies to

\begin{equation}
x_{t+1}^\alpha =\frac{1+\mu }\beta x_t^\alpha (1-x_t)  \label{OG2}
\end{equation}
where $x_t=y/m_t$ and $\alpha =(\sigma -3)/2$, assumed to be positive.
Equation (\ref{OG2}) has a unique positive steady state

\begin{equation}
\bar{x}=1-\frac \beta {1+\mu }  \label{OG3}
\end{equation}
Substituting (\ref{OG3}) into (\ref{OG2}) to eliminate $(1+\mu )/\beta $, we
obtain

\[
x_{t+1}=x_{t}\left( \frac{1-x_{t}}{1-\bar{x}}\right) ^{1/\alpha } 
\]
which for $\alpha =1$ reduces to the logistic map - see Kiminori Matsuyama
(1991) or Azariadis (1993, section 26.4) for more details regarding the
dynamic behavior of this system.

\subsection{Policy Relevancy of Chaos}

As it has just been shown chaos can be produced, for some parameter
settings, from even many of the most classical economic models - including
models in which there is continuous market clearing, rational expectations,
overlapping generations, perfect competition, no externalities, and no forms
of market failure. The issue has been whether or not the parameter settings
that can produce chaos are economically `reasonable'. With large enough
nonlinear, dynamic models to be viewed as possible approximations to
reality, there are no currently available conclusions regarding the
plausibility of the subset of the parameter set that can support chaos.

But there is also the question about whether or not we should care. In
positive economics, there is good reason to care. Understanding the behavior
of an economy that is chaotic is not possible with a model that is not
chaotic, since chaotic solution paths have many properties that cannot be
produced from nonchaotic solutions. But on the normative side, the
usefulness of chaos is much less clear. Grandmont's (1985) model, for
example, produces Pareto optimal chaotic solution paths. The fact that the
solutions are chaotic does not alone provide any justification for
government intervention, and indeed any such intervention could produce a
stable, but Pareto inferior solution. In fact, James Bullard and Alison
Butler (1993) have argued that the existing theoretical results on chaos
have no policy relevance, since in chaotic models the justification for
intervention always can be identified with a form of market failure entered
into the structure of the model, and hence the chaos is an independent and
policy-irrelevant feature of those models.

There is an exemption, however. Woodford (1989) has argued that chaos might
produce increased Pareto-sensitivity to market failure. If that is the case,
then there is an interaction between chaos and the policy implications of
market failure, with small market failures producing increased Pareto loss,
when the economy also is chaotic. This could be an important result and
could result in high policy relevancy for chaos, but at present Woodford's
speculation remains only a supposition, and has not been confirmed in theory
or practice. Hence, at present, the policy relevance of chaos must remain in
doubt.

\section{Efficient Markets and Chaos}

Recently the efficient markets hypothesis and the notions connected with it
have provided the basis for a great deal of research in financial economics.
A voluminous literature has developed supporting this hypothesis. Briefly
stated, the hypothesis claims that asset prices are rationally related to
economic realities and always incorporate all the information available to
the market. This implies the absence of exploitable excess profit
opportunities. However, despite the widespread allegiance to the notion of
market efficiency, a number of studies have suggested that certain asset
prices are not rationally related to economic realities. For example,
Laurence Summers (1986) argues that market valuations differ substantially
and persistently from rational valuations and that existing evidence (based
on common techniques) does not establish that financial markets are
efficient.

Motivated by these considerations, in this section we provide a review of
the literature with respect to the efficient markets hypothesis, discuss
some of the more recent testing methodologies, and finally consider the
intersection between the efficient markets theory and chaos theory.

\subsection{The Random Walk Hypothesis}

Standard asset pricing models typically imply the {\it martingale model},
according to which tomorrow's price is expected to be the same as today's
price. Symbolically, a stochastic process $x_{t}$ follows a martingale if

\begin{equation}
E_{t}(x_{t+1}|\Omega _{t})=x_{t}  \label{Martingale}
\end{equation}
where $\Omega _{t}$ is the time $t$ information set - assumed to include $%
x_{t}$. Equation (\ref{Martingale}) says that if $x_{t}$ follows a
martingale the best forecast of $x_{t+1}$ that could be constructed based on
current information $\Omega _{t}$ would just equal $x_{t}$. Alternatively,
the martingale model implies that $(x_{t+1}-x_{t})$ is a {\it fair game }(a
game which is neither in your favor nor your opponent's)\footnote{%
A stochastic process $z_t$ is a fair game if $z_t$ has the property $%
E_t(z_{t+1}|\Omega _t)=0.$}

\begin{equation}
E_{t}[(x_{t+1}-x_{t})|\Omega _{t}]=0.  \label{Fair game}
\end{equation}
Clearly, $x_{t}$ is a martingale if and only if $(x_{t+1}-x_{t})$ is a fair
game. It is for this reason that fair games are sometimes called {\it %
martingale differences}\footnote{%
The martingale process is a special case of the more general submartingale
process. In particular, $x_{t}$ is a {\it submartingale} if it has the
property $E_{t}(x_{t+1}|\Omega _{t})>x_{t}.$ Note that the submartingale is
also a fair game where $x_{t+1}$ is expected to be greater than $x_{t}.$ In
terms of the $(x_{t+1}-x_{t})$ process the submartingale model implies that $%
E_{t}[(x_{t+1}-x_{t})|\Omega _{t}]>0.$ Stephen LeRoy (1989, pp. 1593-4) also
offers an example in which $E_{t}[(x_{t+1}-x_{t})|\Omega _{t}]<0$, in which
case $x_{t}$ will be a {\it supermartingale}.}.

The fair game model (\ref{Fair game}) says that increments in value (changes
in price adjusted for dividends) are unpredictable, conditional on the
information set $\Omega _{t}$. In this sense, information $\Omega _{t}$ is
fully reflected in prices and hence useless in predicting rates of return.
The hypothesis that prices fully reflect available information has come to
be known as the {\it efficient markets hypothesis}. In fact Eugene Fama
(1970) defined three types of (informational) capital market efficiency (not
to be confused with allocational or Pareto-efficiency), each of which is
based on a different notion of exactly what type of information is
understood to be relevant. In particular, markets are weak-form,
semistrong-form, and strong-form efficient if the information set includes
past prices and returns alone, all public information, and any information
public as well as private, respectively. Clearly, strong-form efficiency
implies semistrong-form efficiency, which in turn implies weak-form
efficiency, but the reverse implications do not follow, since a market
easily could be weak-form efficient but not semistrong-form efficient or
semistrong-form efficient but not strong-form efficient.

The martingale model given by (\ref{Martingale}) can be written equivalently
as

\[
x_{t+1}=x_{t}+\varepsilon _{t} 
\]
where $\varepsilon _{t}$ is the martingale difference. When written in this
form the martingale looks identical to the {\it random walk model} - the
forerunner of the theory of efficient capital markets. The martingale,
however, is less restrictive than the random walk. In particular, the
martingale difference requires only independence of the conditional
expectation of price changes from the available information, as risk
neutrality implies, whereas the (more restrictive) random walk model
requires this and also independence involving the higher conditional moments
(i.e., variance, skewness, and kurtosis) of the probability distribution of
price changes. By not requiring probabilistic independence between
successive price changes, the martingale difference model is entirely
consistent with the fact that price changes, although uncorrelated, tend not
to be independent over time but to have clusters of volatility and
tranquility (i.e., dependence in the higher conditional moments) - a
phenomenon originally noted for stock market prices by Mandelbrot (1963) and
Fama (1965).

\subsection{Tests of the Random Walk Hypothesis}

The random walk and martingale hypotheses imply a unit root in the level of
the price or logarithm of the price series - notice that a unit root is a
necessary but not sufficient condition for the random walk and martingale
models to hold. Hence, these models can be tested using recent advances in
the theory of integrated regressors. The literature on unit root testing is
vast and, in what follows, we shall only briefly illustrate some of the
issues that have arisen in the broader search for unit roots in financial
asset prices.

Charles Nelson and Charles Plosser (1982), using the augmented Dickey-Fuller
(ADF) unit root testing procedure [see David Dickey and Wayne Fuller (1981)]
test the null hypothesis of {\it difference-stationarity} against the {\it %
trend-stationarity} alternative. In particular, in the context of financial
asset prices, one would estimate the following regression

\[
\Delta y_{t}=\alpha _{0}+\alpha _{1}y_{t-1}+\sum_{j=1}^{\ell }c_{j}\Delta
y_{t-j}+\varepsilon _{t} 
\]
where $y$ denotes the logarithm of the series. The null hypothesis of a
single unit root is rejected if $\alpha _{1}$ is negative and significantly
different from zero. A trend variable should not be included, since the
presence of a trend in financial asset prices is a clear violation of market
efficiency, whether or not the asset price has a unit root. The optimal lag
length, $\ell $, can be chosen using data-dependent methods, that have
desirable statistical properties when applied to unit root tests. Based on
such ADF unit root tests, Nelson and Plosser (1982) argue that most
macroeconomic and financial time series have a unit root.

Pierre Perron (1989), however, argues that most time series [and in
particular those used by Nelson and Plosser (1982)] are trend stationary if
one allows for a one-time change in the intercept or in the slope (or both)
of the trend function. The postulate is that certain `big shocks' do not
represent a realization of the underlying data generation mechanism of the
series under consideration and that the null should be tested against the
trend-stationary alternative by allowing, under both the null and the
alternative hypotheses, for the presence of a one-time break (at a known
point in time) in the intercept or in the slope (or both) of the trend
function\footnote{%
Perron's (1989) assumption that the break point is uncorrelated with the
data has been criticized, on the basis that problems associated with
`pre-testing' are applicable to his methodology and that the structural
break should instead be treated as being correlated with the data. More
recently, a number of studies treat the selection of the break point as the
outcome of an estimation procedure and transform Perron's (1989) conditional
(on structural change at a known point in time) unit root test into an
unconditional unit root test.}. Hence, whether the unit root model is
rejected or not depends on how big shocks are treated. If they are treated
like any other shock, then ADF unit root testing procedures are appropriate
and the unit root null hypothesis cannot (in general) be rejected. If,
however, they are treated differently, then Perron-type procedures are
appropriate and the null hypothesis of a unit root will most likely be
rejected.

It is also important to note that in the tests that we discussed so far the
unit root is the null hypothesis to be tested and that the way in which
classical hypothesis testing is carried out ensures that the null hypothesis
is accepted unless there is strong evidence against it. In fact, Denis
Kwiatkowski, Peter Phillips, Peter Schmidt, and Yongcheol Shin (1992) argue
that such unit root tests fail to reject a unit root because they have low
power against relevant alternatives and they propose tests (known as KPSS
tests) of the hypothesis of stationarity against the alternative of a unit
root. They argue that such tests should complement unit root tests and that
by testing both the unit root hypothesis and the stationarity hypothesis,
one can distinguish series that appear to be stationary, series that appear
to be integrated, and series that are not very informative about whether or
not they are stationary or have a unit root.

Finally, given that integration tests are sensitive to the class of models
considered (and may be misleading because of misspecification), {\it %
fractionally}-integrated representations, which nest the unit-root
phenomenon in a more general model, have also been used - see Richard
Baillie (1996) for a survey. Fractional integration is a popular way to
parameterize long-memory processes. If such processes are estimated with the
usual autoregressive-moving average model, without considering fractional
orders of integration, the estimated autoregressive process can exhibit
spuriously high persistence close to a unit root. Since financial asset
prices might depart from their means with long memory, one could condition
the unit root tests on the alternative of a fractional integrated process,
rather than the usual alternative of the series being stationary. In this
case, if we fail to reject an autoregressive unit root, we know it is not a
spurious finding due to neglect of the relevant alternative of fractional
integration and long memory.

Despite the fact that the random walk and martingale hypotheses are
contained in the null hypothesis of a unit root, unit root tests are not
predictability tests. They are designed to reveal whether a series is
difference-stationary or trend stationary and as such they are tests of the
permanent/temporary nature of shocks. More recently a series of papers
including those by James Poterba and Summers (1988), and Andrew Lo and Craig
MacKinlay (1988) have argued that the efficient markets theory can be tested
by comparing the relative variability of returns over different horizons
using the variance ratio methodology of John Cochrane (1988). They have
shown that asset prices are mean reverting over long investment horizons -
that is, a given price change tends to be reversed over the next several
years by a predictable change in the opposite direction. Similar results
have been obtained by Fama and Kenneth French (1988), using an alternative
but closely related test based on predictability of multiperiod returns. Of
course, mean-reverting behavior in asset prices is consistent with
transitory deviations from equilibrium which are both large and persistent,
and implies positive autocorrelation in returns over short horizons and
negative autocorrelation over longer horizons.

Predictability of financial asset returns is a broad and very active
research topic and a complete survey of the vast literature is beyond the
scope of the present paper. We shall notice, however, that a general
consensus has emerged that asset returns are predictable. As John Campbell,
Lo, and MacKinlay (1997, pp. 80) put it ``[r]ecent econometric advances and
empirical evidence seem to suggest that financial asset returns are
predictable to some degree. Thirty years ago this would have been tantamount
to an outright rejection of market efficiency. However, modern financial
economics teaches us that other, perfectly rational, factors may account for
such predictability. The fine structure of securities markets and frictions
in the trading process can generate predictability. Time-varying expected
returns due to changing business conditions can generate predictability. A
certain degree of predictability may be necessary to reward investors for
bearing certain dynamic risks''.

\subsection{Random Walk versus Chaos}

Most of the empirical tests that we discussed in the previous subsection are
designed to detect `linear' structure in financial data - that is, linear
predictability is the focus. However, as Campbell, Lo, and MacKinlay (1997,
pp. 467) argue ``many aspects of economic behavior may not be linear.
Experimental evidence and casual introspection suggest that investors'
attitudes towards risk and expected return are nonlinear. The terms of many
financial contracts such as options and other derivative securities are
nonlinear. And the strategic interactions among market participants, the
process by which information is incorporated into security prices, and the
dynamics of economy-wide fluctuations are all inherently nonlinear.
Therefore, a natural frontier for financial econometrics is the modeling of
nonlinear phenomena''.

It is for this reason that interest in deterministic nonlinear chaotic
processes has in the recent past experienced a tremendous rate of
development. Besides its obvious intellectual appeal, chaos is interesting
because of its ability to generate output that mimics the output of
stochastic systems thereby offering an alternative explanation for the
behavior of asset prices. In fact, the possible existence of chaos could be
exploitable and even invaluable. If, for example, chaos can be shown to
exist in asset prices, the implication would be that profitable,
nonlinearity-based trading rules exist (at least in the short run and
provided the actual generating mechanism is known). Prediction, however,
over long periods is all but impossible, due to the sensitive dependence on
initial conditions property of chaos.

Clearly then, an important area for potentially productive research is to
test for chaos and (in the event that it exists) to identify the nonlinear
deterministic system that generates it. We turn to such tests in the
following section.

\section{Tests of Nonlinearity and Chaos}

Although the exciting recent advances in deterministic nonlinear dynamical
systems theory have had immediate implications for the `hard' sciences, the
impact on economics and finance has been less dramatic for at least two
reasons. First, unlike most hard scientists, economists are generally not
specific about functional form when modeling economic phenomena as
deterministic nonlinear dynamical systems. Thus they rarely have theoretical
reasons for expecting to find one form of nonlinearity rather than another.
Second, economists mostly use non-experimental data, rendering it almost
impossible to recover the deterministic dynamical system governing economic
phenomena, even if such a system exists and is low-dimensional.

Despite these caveats, the mathematics of deterministic nonlinear dynamical
systems has motivated several univariate statistical tests for independence,
nonlinearity, and chaos, to which we now turn.

\subsection{The Correlation Dimension Test}

The concept and measurement of fractal dimension are not only necessary to
understand the finer geometrical nature of strange attractors, but they are
also fundamental tools for providing quantitative analyses of such
attractors. Unfortunately, however, fractal dimension [as defined by
Equation ($4$)] cannot be computed easily in practice, and convergence of
the limit may not be guaranteed. To remedy this, Peter Grassberger and
Itamar Procaccia (1983) suggested the concept of {\it correlation dimension}
(or {\it correlation exponent}) which is, at the moment, prevailing in
applications. The basic idea is that of replacing the box-counting
algorithm, necessary to compute $N(\epsilon )$ in Equation (4), with the
measurement of correlations between points of a long time series on the
attractor. Hence, the correlation dimension (unlike the fractal dimension)
is a probabilistic, not a metric, dimension.

To briefly discuss the correlation dimension test for chaos, let us start
with the $1$-dimensional series, $\left\{ x_{t}\right\} _{t=1}^{n}$, which
can be embedded into a series of $m$-dimensional vectors $%
X_{t}=(x_{t},x_{t-1},...,x_{t-m+1})^{\prime }$ giving the series $\left\{
x_{t}\right\} _{t=m}^{n}$. The selected value of $m$ is called the {\it %
embedding dimension} and each $X_{t}$ is known as an $m$-{\it history} of
the series $\left\{ x_{t}\right\} _{t=1}^{n}$. This converts the series of
scalars into a slightly shorter series of ($m$-dimensional) vectors with
overlapping entries\footnote{%
In creating the embedding one could also take a forward orientation, $%
X_{t}=(x_{t},x_{t+1},...,x_{t+m-1})^{\prime }$, without affecting the
results.}. In particular, from the sample size $n,$ $N=n-m+1$ $m$-histories
can be made\footnote{%
For example, the series $\left\{ x_{1},...,x_{6}\right\} $ would give the
following four overlapping $3$-histories: $X_{3}=(x_{1},x_{2},x_{3})^{\prime
},$ $X_{4}=(x_{2},x_{3},x_{4})^{\prime },$ $X_{5}=(x_{3},x_{4},x_{5})^{%
\prime },$ and $X_{6}=(x_{4},x_{5},x_{6})^{\prime }.$}. Assuming that the
true, but unknown, system which generated $\left\{ x_{t}\right\} _{t=1}^{n}$
is $\vartheta $-dimensional and provided that $m\geq 2\vartheta +1$, then
the $N$ $m$-histories recreate the dynamics of the data generation process
and can be used to analyze the dynamics of the system.

The correlation dimension test is based on the {\it correlation function}
(or {\it correlation} {\it integral}), $C(N,m,\epsilon )$, which for a given
embedding dimension $m$ is given by:

\[
C(N,m,\epsilon )=\frac{1}{N(N-1)}\sum_{m\leq t\neq s\leq N}H\left( \epsilon
-\left\| X_{t}-X_{s}\right\| \right) 
\]
where $\epsilon $ is a sufficiently small number, $H(z)$ is the Heavside
function, which maps positive arguments into $1$, and nonpositive arguments
into $0$, i.e.,

\[
H(z)=\left\{ 
\begin{array}{cc}
1 & \text{if }z>0 \\ 
0 & \text{otherwise,}
\end{array}
\right. 
\]
and $\left\| .\right\| $ denotes the distance induced by the selected norm%
\footnote{%
Brock (1986, Theorem 2.4) shows that the correlation integral is independent
of the choice of norm. The type most often used is the maximum norm (which
is also more convenient for computer applications): $\left\|
X_{t}-X_{s}\right\| =\max_{k\in \left[ 0,m-1\right] }\left\{ \left|
x_{t+k}-x_{s+k}\right| \right\} $, where$\left\| .\right\| $ is Euclidean
distance. Using this norm the correlation integral may be written as $%
C(N,m,\epsilon )=\frac{1}{N(N-1)}\sum_{m\leq t\neq s\leq
N}\prod_{k=0}^{m-1}H\left( \epsilon -\left| X_{t+k}-X_{s+k}\right| \right) $
since $H\left( \epsilon -\left\| X_{t}-X_{s}\right\| \right)
=\prod_{k=0}^{m-1}H\left( \epsilon -\left| X_{t+k}-X_{s+k}\right| \right) $,
i.e., if any $\left| x_{t+k}-x_{s+k}\right| \geq \epsilon $ then $H(.)=0.$}.
In other words, the correlation integral is the number of pairs $(t,s)$ such
that each corresponding component of $X_{t}$ and $X_{s}$ are near to each
other, nearness being measured in terms of distance being less than $%
\epsilon $. Intuitively, $C(N,m,\epsilon )$ measures the probability that
the distance between any two $m$-histories is less than $\epsilon $. If $%
C(N,m,\epsilon )$ is large (which means close to $1$) for a very small $%
\varepsilon $, then the data is very well correlated.

To move from the correlation function to the correlation dimension, one
proceeds by looking to see how $C(N,m,\epsilon )$ changes as $\epsilon $
changes. One expects $C(N,m,\epsilon )$ to increase with $\epsilon $ (since
increasing $\epsilon $ increases the number of neighbouring points that get
included in the correlation integral). In fact, Grassberger and Procaccia
(1983) have shown that for small values of $\epsilon $, $C(N,m,\epsilon )$
grows exponentially at the rate of $D_{c}$

\[
C(N,m,\epsilon )=\eta e^{D_{c}} 
\]
where $\eta $ is some constant and $D_{c}$ is the above mentioned
correlation dimension.

If the increase in $C(N,m,\epsilon )$ is slow as $\epsilon $ is increased,
then most data points have to be near to each other, and the data is well
correlated. If, however, the increase is fast, then the data are rather
uncorrelated. Hence, the higher the correlation dimension [and the faster
the increase in $C(N,m,\epsilon )$ as $\epsilon $ is increased], the less
correlated the data is and the system is regarded stochastic. On the other
hand, the lower the correlation dimension [and the slower the increase in $%
C(N,m,\epsilon )$ as $\epsilon $ is increased], the more correlated the data
is and the system is regarded as essentially deterministic, even if fairly
complicated.

The correlation dimension can be defined as

\[
D_{c}=\lim_{\epsilon \rightarrow 0}\frac{d\log C(N,m,\epsilon )}{d\log
\epsilon } 
\]
that is, by the slope of the regression of $\log C(N,m,\epsilon )$ versus $%
\log \epsilon $ for small values of $\epsilon $. As a practical matter one
investigates the estimated value of $D_{c}$ as $m$ is increased. If as $m$
increases $D_{c}$ continues to rise, then the system is stochastic. If,
however, the data are generated by a deterministic process (consistent with
chaotic behavior), then $D_{c}$ reaches a finite saturation limit beyond
some relatively small $m\footnote{%
Since $D_{c}$ can be used to characterize both chaos and stochastic dynamics
(i.e., $D_{c}$ is a finite number in the case of chaos and equal to infinity
in the case of an independent and identically distributed stochastic
process), one often finds in the literature expressions like `deterministic
chaos' (meaning simply chaos) and `stochastic chaos' (meaning standard
stochastic dynamics). This terminology, however, is confusing in contexts
other than that of the correlation dimension analysis and we shall not use
it in this paper.}$. The correlation dimension can therefore be used to
distinguish true stochastic processes from deterministic chaos (which may be
low-dimensional or high-dimensional)\footnote{%
It is to be noted that Grassberger and Procaccia (1983) have shown that $%
D_c\leq D$, i.e., $D_c$ is a lower bound for $D$ - see also Medio (1992) for
other measures of fractal dimension and their relation to $D_c$ and $D$.}.

While the correlation dimension measure is therefore potentially very useful
in testing for chaos, the sampling properties of the correlation dimension
are, however, unknown. As William Barnett, Ronald Gallant, Melvin Hinich,
Jochen Jungeilges, Daniel Kaplan, and Mark Jensen (1995, pp. 306) put it
``[i]f the only source of stochasticity is noise in the data, and if that
noise is slight, then it is possible to filter the noise out of the data and
use the correlation dimension test deterministically. However, if the
economic structure that generated the data contains a stochastic disturbance
within its equations, the correlation dimension is stochastic and its
derived distribution is important in producing reliable inference''.

Moreover, if the correlation dimension is very large as in the case of
high-dimensional chaos, it will be very difficult to estimate it without an
enormous amount of data. In this regard, Ruelle (1990) argues that a chaotic
series can only be distinguished if it has a correlation dimension well
below $2\log _{10}N$, where $N$ is the size of the data set, suggesting that
with economic time series the correlation dimension can only distinguish
low-dimensional chaos from high-dimensional stochastic processes - see also
Grassberger and Procaccia (1983) for more details\footnote{%
Therefore, to detect an attractor with $D_c=2$ we need at least $460$ data
points, with $D_c=3$ at least $10,000$ data points, and with $D_c=4$ at
least $210,000$ data points.}.

\subsection{The BDS Test}

To deal with the problems of using the correlation dimension test, Brock,
Davis Dechert, Blake LeBaron, and Jos\'{e} Scheinkman (1996) devised a new
statistical test which is known as the BDS test. The BDS tests the null
hypothesis of whiteness (independent and identically distributed
observations) against an unspecified alternative using a nonparametric
technique.

The BDS test is based on the Grassberger and Procaccia (1983) correlation
integral as the test statistic. In particular, under the null hypothesis of
whiteness, the BDS statistic is

\[
W(N,m,\epsilon )=\sqrt{N}\frac{C(N,m,\epsilon )-C(N,1,\epsilon )^{m}}{%
\widehat{\sigma }(N,m,\epsilon )} 
\]
where $\widehat{\sigma }(N,m,\epsilon )$ is an estimate of the asymptotic
standard deviation of $C(N,m,\epsilon )-C(N,1,\epsilon )^{m}$ - the formula
for $\widehat{\sigma }(N,m,\epsilon )$ can be found in Brock et al. (1996).
The BDS statistic is asymptotically standard normal under the whiteness null
hypothesis - see Brock et al. (1996) for details.

The intuition behind the BDS statistic is as follows. $C(N,m,\epsilon )$ is
an estimate of the probability that the distance between any two $m$%
-histories, $X_{t}$ and $X_{s}$ of the series $\left\{ x_{t}\right\} $ is
less than $\epsilon $. If $\left\{ x_{t}\right\} $ were independent then for 
$t\neq s$ the probability of this joint event equals the product of the
individual probabilities. Moreover, if $\left\{ x_{t}\right\} $ were also
identically distributed then all of the $m$ probabilities under the product
sign are the same. The BDS statistic therefore tests the null hypothesis
that $C(N,m,\epsilon )=C(N,1,\epsilon )^{m}$ - the null hypothesis of
whiteness\footnote{%
Note that whiteness implies that $C(N,m,\epsilon )=C(N,1,\epsilon )^{m}$ but
the converse is not true.}.

Since the asymptotic distribution of the BDS test statistic is known under
the null hypothesis of whiteness, the BDS test provides a direct (formal)
statistical test for whiteness against general dependence, which includes
both nonwhite linear and nonwhite nonlinear dependence. Hence, the BDS test
does not provide a direct test for nonlinearity or for chaos, since the
sampling distribution of the test statistic is not known (either in finite
samples or asymptotically) under the null hypothesis of nonlinearity,
linearity, or chaos. It is, however, possible to use the BDS test to produce
indirect evidence about nonlinear dependence [whether chaotic (i.e.,
nonlinear deterministic) or stochastic], which is necessary but not
sufficient for chaos - see Barnett et al. (1997) and Barnett and Melvin
Hinich (1992) for a discussion of these issues.

\subsection{The Hinich Bispectrum Test}

Hinich (1982) argues that the bispectrum in the frequency domain is easier
to interpret than the multiplicity of third order moments $\left\{
C_{xxx}(r,s):s\leq r,r=0,1,2,...\right\} $ in the time domain. For
frequencies $\omega _{1}$ and $\omega _{2}$ in the principal domain given by

\[
\Omega =\left\{ (\omega _1,\omega _2)\text{ : }0<\omega _1<0.5,\omega
_2<\omega _1,2\omega _1+\omega _2<1\right\} , 
\]
the bispectrum, $B_{xxx}(\omega _1,\omega _2)$, is defined by

\[
B_{xxx}(\omega _1,\omega _2)=\sum_{r=-\infty }^\infty \sum_{s=-\infty
}^\infty C_{xxx}(r,s)\exp \left[ -i2\pi (\omega _1r+\omega _2s)\right] . 
\]
The bispectrum is the double Fourier transformation of the third order
moments function and is the third order polyspectrum. The regular power
spectrum is the second order polyspectrum and is a function of only one
frequency.

The skewness function $\Gamma (\omega _{1},\omega _{2})$ is defined in terms
of the bispectrum as follows

\begin{equation}
\Gamma ^{2}(\omega _{1},\omega _{2})=\frac{|B_{xxx}(\omega _{1},\omega
_{2})|^{2}}{S_{xx}(\omega _{1})S_{xx}(\omega _{2})S_{xx}(\omega _{1}+\omega
_{2})},  \label{Skewness}
\end{equation}
where $S_{xx}(\omega )$ is the (ordinary power) spectrum of $x(t)$ at
frequency $\omega $. Since the bispectrum is complex valued, the absolute
value (vertical lines) in Equation (\ref{Skewness}) designates modulus.
David Brillinger (1965) proves that the skewness function $\Gamma (\omega
_{1},\omega _{2})$ is constant over all frequencies $(\omega _{1},\omega
_{2})\in \Omega $ if $\left\{ x(t)\right\} $ is linear; while $\Gamma
(\omega _{1},\omega _{2})$ is flat at zero over all frequencies if $\left\{
x(t)\right\} $ is Gaussian. Linearity and Gaussianity can be tested using a
sample estimator of the skewness function. But observe that those flatness
conditions are necessary but not sufficient for general linearity and
Gaussianity, respectively. On the other hand, flatness of the skewness
function is necessary and sufficient for third order nonlinear dependence.
The Hinich (1982) `linearity test' tests the null hypothesis that the
skewness function is flat, and hence is a test of lack of third order
nonlinear dependence. For details of the test, see Hinich (1982).

\subsection{The NEGM Test}

As it was argued earlier, the distinctive feature of chaotic systems is
sensitive dependence on initial conditions - that is, exponential divergence
of trajectories with similar initial conditions. The most important tool for
diagnosing the presence of sensitive dependence on initial conditions (and
thereby of chaoticity) is provided by the dominant Lyapunov exponent, $%
\lambda $. This exponent measures average exponential divergence or
convergence between trajectories that differ only in having an
`infinitesimally small' difference in their initial conditions and remains
well-defined for noisy systems. A bounded system with a positive Lyapunov
exponent is one operational definition of chaotic behavior.

One early method for calculating the dominant Lyapunov exponent is that
proposed by Alan Wolf, Jack Swift, Harry Swinney, and John Vastano (1985).
This method, however, requires long data series and is sensitive to dynamic
noise, so inflated estimates of the dominant Lyapunov exponent are obtained.
Recently, Douglas Nychka, Stephen Ellner, Ronald Gallant, and Daniel
McCaffrey (1992) have proposed a regression method, involving the use of
neural network models, to test for positivity of the dominant Lyapunov
exponent. The Nychka et al. (1992), hereafter NEGM, Lyapunov exponent
estimator is a regression (or Jacobian) method, unlike the Wolf et al.
(1985) direct method which [as Brock and Chera Sayers (1988) have found]
requires long data series and is sensitive to dynamic noise.

Assume that the data $\{x_t\}$ are real-valued and are generated by a
nonlinear autoregressive model of the form

\begin{equation}
x_{t}=f(x_{t-L},x_{t-2L},...,x_{t-mL})+e_{t}  \label{Autoregressive}
\end{equation}
for $1\leq t\leq N$, where $L$ is the time-delay parameter and $m$ is the
length of the autoregression. Here $f$ is a smooth unknown function, and $%
\{e_{t}\}$ is a sequence of independent random variables with zero mean and
unknown constant variance. The Nychka et al. (1992) approach to estimation
of the maximum Lyapunov exponent involves producing a state-space
representation of (\ref{Autoregressive})

\[
X_{t}=F(X_{t-L})+E_{t},\quad F\text{ : }{\Bbb R}^{m}\rightarrow {\Bbb R}^{m} 
\]
where $X_{t}=(x_{t},x_{t-L},...,x_{t-mL+L})^{\prime }$, $%
F(X_{t-L})=(f(x_{t-L},...,x_{t-mL}),x_{t-L},...,$ $x_{t-mL+L})^{\prime }$,
and $E_{t}=(e_{t},0,...,0)^{\prime }$, and using a Jacobian-based method to
estimate $\lambda $ through the intermediate step of estimating the
individual Jacobian matrices

\[
J_t=\frac{\partial F(X_t)}{\partial X^{\prime }}. 
\]

After using several nonparametric methods, McCaffrey et al. (1992) recommend
using either thin plate splines or neural nets to estimate $J_t.$ Estimation
based on neural nets involves the use of the a neural net with $q$ units in
the hidden layer

\[
f(X_{t-L},\theta )=\beta _{0}+\sum_{j=1}^{q}\beta _{j}\psi (\gamma
_{0j}+\sum_{i=1}^{m}\gamma _{ij}x_{t-iL}) 
\]
where $\psi $ is a known (hidden) nonlinear `activation function' [usually
the logistic distribution function $\psi (u)=1/(1+\exp (-u))]$. The
parameter vector $\theta $ is then fit to the data by nonlinear least
squares. That is, one computes the estimate $\widehat{\theta }$ to minimize
the sum of squares $S(\theta )=\sum_{t=1}^{N}\left[ x_{t}-f(X_{t-1},\theta
)\right] ^{2}$, and uses $\widehat{F}(X_{t})=(f(x_{t-L},...,x_{t-mL},%
\widehat{\theta }),x_{t-L},...,x_{t-mL+L})^{\prime }$ to approximate $%
F(X_{t})$.

As appropriate values of $L,m,$ and $q$, are unknown, Nychka et al. (1992)
recommend selecting that value of the triple $(L,m,q)$ that minimizes the
Bayesian Information Criterion (BIC) - see Gideon Schwartz (1978). As shown
by Gallant and Halbert White (1992), we can use $\widehat{J}_{t}=\partial 
\widehat{F}(X_{t})/\partial X^{\prime }$ as a nonparametric estimator of $%
J_{t}$ when $(L,m,q)$ are selected to minimize BIC. The estimate of the
dominant Lyapunov exponent then is

\[
\widehat{\lambda }=\frac{1}{2N}\log \left| \widehat{v}_{1}(N)\right| 
\]
where $\widehat{v}_{1}(N)$ is the largest eigenvalue of the matrix $\widehat{%
T}_{N}^{\prime }\widehat{T}_{N}$ and where $\widehat{T}_{N}=\widehat{J}_{N}%
\widehat{J}_{N-1},...,\widehat{J}_{1}$.

\subsection{The White Test}

In White's (1989) test, the time series is fitted by a single hidden-layer
feed-forward neural network, which is used to determine whether any
nonlinear structure remains in the residuals of an autoregressive (AR)
process fitted to the same time series. The null hypothesis for the test is
`linearity in the mean' relative to an information set. A process that is
linear in the mean has a conditional mean function that is a linear function
of the elements of the information set, which usually contains lagged
observations on the process\footnote{%
For a formal definition of linearity in the mean, see Tae-Hwy Lee, White,
and Granger (1993, section 1). Note that a process that is not linear in the
mean is said to exhibit `neglected nonlinearity'. Also, a process that is
linear is also linear in the mean, but the converse need not be true.}.

The rationale for White's test can be summarized as follows: under the null
hypothesis of linearity in the mean, the residuals obtained by applying a
linear filter to the process should not be correlated with any measurable
function of the history of the process. White's test uses a fitted neural
net to produce the measurable function of the process's history and an AR
process as the linear filter. White's method then tests the hypothesis that
the fitted function does not correlate with the residuals of the AR process.
The resulting test statistic has an asymptotic $\chi ^{2}$ distribution
under the null of linearity in the mean\footnote{%
See Lee, White, and Granger (1993, section 2) for a presentation of the test
statistic's formula and computation method.}.

\subsection{The Kaplan Test}

We begin our discussion of Daniel Kaplan's (1994) test by reviewing its
origins in the chaos literature, although the test is currently being used
as a test of linear stochastic process against general nonlinearity, whether
or not noisy or chaotic. In the case of chaos, a time series plot of the
output of a chaotic system may be very difficult to distinguish visually
from a stochastic process. However, plots of the solution paths in phase
space ($x_{t+1}$ plotted against $x_{t}$ and lagged values of $x_{t}$) often
reveal deterministic structure that was not evident in a plot of $x_{t}$
versus $t$ - see, for example, Figure $9$. A test based upon continuity in
phase space has been proposed by Kaplan (1994).

Briefly, he used the fact that deterministic solution paths, unlike
stochastic processes, have the following property: points that are nearby
are also nearby under their image in phase space. Using this fact, he has
produced a test statistic, which has a strictly positive lower bound for a
stochastic process, but not for a deterministic solution path. By computing
the test statistic from an adequately large number of linear processes that
plausibly might have produced the data, the approach can be used to test for
linearity against the alternative of noisy nonlinear dynamics. The procedure
involves producing linear stochastic process surrogates for the data and
determining whether the surrogates or a noisy continuous nonlinear dynamical
solution path better describe the data. Linearity is rejected, if the value
of the test statistic from the surrogates is never small enough relative to
the value of the statistic computed from the data\footnote{%
See Kaplan (1994) or Barnett et al. (1997) for more details about Kaplan's
procedure.}.

\section{Evidence on Nonlinearity and Chaos}

There have been a great deal of studies over the past few years testing for
nonlinearity or chaos on economic and financial data. Thus we devote a good
deal of space to this empirical work. In this section we present a
discussion of the empirical evidence on economic and financial data, look at
the controversies that have arisen about the available results, address one
important question regarding the power of some of the best known tests for
nonlinearity or chaos against various alternatives, and raise the issue of
whether dynamical systems theory is practical in economics.

\subsection{Evidence on Economic Data}

In Table 1 we list 7 studies that have used various economic time series to
test for nonlinearity or chaos. In Columns 2 to 5 we present the data set;
the number of observations; the testing procedure used; and the results
obtained. Clearly, there is a broad consensus of support for the proposition
that the data generating processes are characterized by a pattern of
nonlinear dependence, but there is no consensus at all on whether there is
chaos in economic time series. For example, Brock and Sayers (1988), Murray
Frank and Thanasis Stengos (1988), and Frank, Ramazan Gencay, and Stengos
(1988) find no evidence of chaos in U.S., Canadian, and international,
respectively, macroeconomic time series.

On the other hand, Barnett and Ping Chen (1988), claimed successful
detection of chaos in the (demand-side) U.S. Divisia monetary aggregates.
Their conclusion was further confirmed by Gregory DeCoster and Douglas
Mitchell (1991, 1994). This published claim of successful detection of chaos
has generated considerable controversy, as in James Ramsey, Sayers, and
Philip Rothman (1990) and Ramsey and Rothman (1994), who by re-examining the
data utilized in Barnett and Chen (1988) show that there is no evidence for
the presence of chaos. In fact, they raised similar questions regarding
virtually all of the other published tests of chaos.

Further results relevant to this controversy have recently been provided by
Apostolos Serletis (1995). Building on Barnett and Chen (1988), Serletis
(1995) contrasts the random walk behavior of the velocity of money to
chaotic dynamics, motivated by the notion that velocity follows a
deterministic, dynamic, and nonlinear process which generates output that
mimics the output of stochastic systems. In doing so, he tests for chaos
using the Lyapunov exponent estimator of Nychka et al. (1992) and reports
evidence of chaos in the Divisia L velocity series.

Although from a theoretical point of view, it would be extremely interesting
to obtain empirical verification that macroeconomic series have actually
been generated by deterministic chaotic systems, it is fair to say that
those series are not the most suitable ones for the calculation of chaos
indicators. This is for at least two reasons. First of all, the series are
rather short with regard to the calculations to be performed, since they are
usually recorded at best only monthly; secondly, they have probably been
contaminated by a substantial dose of noise (this is particularly true for
aggregate time series like GNP). We should not be surprised, therefore, that
exercises of this kind have not yet led to particularly encouraging results.

\subsection{Evidence on Financial Data}

As can be seen from Table 2 (where we summarize the evidence in the same
fashion as in Table 1), there is already a substantial literature testing
for nonlinear dynamics on financial data, using various inference methods -
for other unpublished work on testing nonlinearity and chaos on financial
data, see Abhay Abhyankar, Laurence Copeland, and Woon Wong (1997, table 1).
In fact, the analysis of financial time series has led to results which are
as a whole more interesting and more reliable than those of macroeconomic
series. This is probably due to the much larger number of data available and
their superior quality (measurement in most cases is more precise, at least
when we do not have to make recourse to broad aggregation).

Scheinkman and LeBaron (1989) studied United States weekly returns on the
Center for Research in Security Prices (CRSP) value-weighted index,
employing the BDS statistic, and found rather strong evidence of
nonlinearity and some evidence of chaos\footnote{%
In order to verify the presence of a nonlinear structure in the data, they
also suggested employing the so-called `shuffling diagnostic'. This
procedure involves studying the residuals obtained by adapting an
autoregressive model to a series and then reshuffling these residuals. If
the residuals are totally random (i.e., if the series under scrutiny is not
characterized by chaos), the dimension of the residuals and that of the
shuffled residuals should be approximately equal. On the contrary, if the
residuals are chaotic and have some structure, then the reshuffling must
reduce or eliminate the structure and consequently increase the correlation
dimension. The correlation dimension of their reshuffled residuals always
appeared to be much greater than that of the original residuals, which was
interpreted as being consistent with chaos.}. Some very similar results have
been obtained by Frank and Stengos (1989), investigating daily prices (from
the mid 1970's to the mid 1980's) for gold and silver, using the correlation
dimension and the Kolmogorov entropy. Their estimate of the correlation
dimension was between 6 and 7 for the original series and much greater and
non-converging for the reshuffled data.

More recently, Serletis and Periklis Gogas (1997) test for chaos in seven
East European black market exchange rates, using the Kees Koedijk and
Clements Kool (1992) monthly data (from January 1955 through May 1990). In
doing so, they use three inference methods, the BDS test, the NEGM test, as
well as the Lyapunov exponent estimator of Gencay and Dechert (1992). They
find some consistency in inference across methods, and conclude, based on
the NEGM test, that there is evidence consistent with a chaotic nonlinear
generation process in two out of the seven series - the Russian ruble and
East German mark. Altogether, these and similar results seem to suggest that
financial series provide a more promising field of research for the methods
in question.

A notable feature of the literature just summarized is that most
researchers, in order to find sufficient observations to implement the
tests, use data periods measured in years. The longer the data period,
however, the less plausible is the assumption that the underlying data
generation process has remained stationary, thereby making the results
difficult to interpret. In fact, different conclusions have been reached by
researchers using high-frequency data over short periods. For example,
Abhyankar, Copeland, and Wong (1995) examine the behavior of the U.K.
Financial Times Stock Exchange 100 (FTSE 100) index, over the first six
months of 1993 (using 1-, 5-, 15-, 30-, and 60-minute returns). Using the
Hinich (1982) bispectral linearity test, the BDS test, and the NEGM\ test,
they find evidence of nonlinearity, but no evidence of chaos.

More recently, Abhyankar, Copeland, and Wong (1997) test for nonlinear
dependence and chaos in real-time returns on the world's four most important
stock-market indices - the FTSE 100, the Standard \& Poor 500 (S\&P 500)
index, the Deutscher Aktienindex (DAX), and the Nikkei 225 Stock Average.
Using the BDS and the NEGM tests, and 15-second, 1-minute, and 5-minute
returns (from September 1 to November 30, 1991), they reject the hypothesis
of independence in favor of a nonlinear structure for all data series, but
find no evidence of low-dimensional chaotic processes.

Of course, there is other work, using high-frequency data over short
periods, that finds order in the apparent chaos of financial markets. For
example, the article by Shoaleh Ghashghaie, Wolfgang Breymann, Joachim
Peinke, Peter Talkner, and Yadolah Dodge (1996) analyzes all worldwide
1,472,241 bid-ask quotes on U.S. dollar-German mark exchange rates between
October 1, 1992 and September 30, 1993. It applies physical principles and
provides a mathematical explanation of how one trading pattern led into and
then influenced another. As the authors conclude, ``...we have reason to
believe that the qualitative picture of turbulence that has developed during
the past 70 years will help our understanding of the apparently remote field
of financial markets''.

\subsection{Controversies}

As discussed in the previous two subsections, there is little agreement
about the existence of chaos or even of nonlinearity in economic and
financial data, and some economists continue to insist that linearity
remains a good assumption for such data, despite the fact that theory
provides very little support for that assumption. It should be noted,
however, that the available tests search for evidence of nonlinearity or
chaos in data without restricting the boundary of the system that could have
produced that nonlinearity or chaos. Hence these tests should reject
linearity, even if the structure of the economy is linear, but the economy
is subject to shocks from a surrounding nonlinear or chaotic physical
environment, as through nonlinear climatological or weather dynamics. Under
such circumstances, linearity would seem an unlikely inference\footnote{%
In other words, not only is there no reason in economic theory to expect
linearity within the structure of the economy, but there is even less reason
to expect to find linearity in nature, which produces shocks to the system.}.

Since the available tests are not structural and hence have no ability to
identify the source of detected chaos, the alternative hypothesis of the
available tests is that no natural deterministic explanation exists for the
observed economic fluctuations anywhere in the universe. In other words, the
alternative hypothesis is that economic fluctuations are produced by
supernatural shocks or by inherent randomness in the sense of quantum
physics. Considering the implausibility of the alternative hypothesis, one
would think that findings of chaos in such nonparametric tests would produce
little controversy, while any claims to the contrary would be subjected to
careful examination. Yet, in fact the opposite seems to be the case.

We argued earlier that the controversies might stem from the high noise
level that exists in most aggregated economic time series and the relatively
low sample sizes that are available with economic data. However, it also
appears that the controversies are produced by the nature of the tests
themselves, rather than by the nature of the hypothesis, since linearity is
a very strong null hypothesis, and hence should be easy to reject with any
test and any economic or financial time series on which an adequate sample
size is available. In particular, there may be very little robustness of
such tests across variations in sample size, test method, and data
aggregation method. That possibility was the subject of Barnett et al.
(1995), who used five of the most widely used tests for nonlinearity or
chaos with various monetary aggregate data series of various sample sizes
and acquired results that differed substantially across tests and over
sample sizes, as well as over the statistical index number formulas used to
aggregate over the same component data.

\subsection{Single Blind Controlled Competition}

It is possible that none of the tests for chaos and nonlinear dynamics that
we have discussed completely dominates the others, since some tests may have
higher power against certain alternatives than other tests, without any of
the tests necessarily having higher power against all alternatives. If this
is the case, each of the tests may have its own comparative advantages, and
there may even be a gain from using more than one of the tests in a sequence
designed to narrow down the alternatives.

To explore this possibility, Barnett with the assistance of Jensen designed
and ran a single blind controlled experiment, in which they produced
simulated data from various processes having linear, nonlinear chaotic, or
nonlinear nonchaotic signal. They transmitted each simulated data set by
e-mail to experts in running each of the statistical tests that were entered
into the competition. The e-mailed data included no identification of the
generating process, so those individuals who ran the tests had no way of
knowing the nature of the data generating process, other than the sample
size, and there were two sample sizes: a `small sample' size of 380 and a
`large sample' size of 2000 observations.

In fact five generating models were used to produce samples of the small and
large size. The models were a fully deterministic, chaotic Feigenbaum
recursion (Model I), a generalized autoregressive conditional
heteroskedasticity (GARCH) process (Model II), a nonlinear moving average
process (Model III), an autoregressive conditional heteroskedasticity (ARCH)
process (Model IV), and an autoregressive moving average (ARMA)\ process
(Model V). Details of the parameter settings and noise generation method can
be found in Barnett et al. (1996). The tests entered into this competition
were Hinich's bispectrum test, the BDS test, White's test, Kaplan's test,
and the NEGM\ test of chaos.

The results of the competition are available in Barnett et al. (1997) and
are summarized in Table $3$. They provide the most systematic available
comparison of tests of nonlinearity and indeed do suggest differing powers
of each test against certain alternative hypotheses. In comparing the
results of the tests, however, one factor seemed to be especially important:
subtle differences existed in the definition of the null hypothesis, with
some of the tests being tests of the null of linearity, defined in three
different manners in the derivation of the test's properties, and one test
being a test of the null of chaos. Hence there were four null hypotheses
that had to be considered to be able to compare each test's power relative
to each test's own definition of the null.

Since the tests do not all have the same null hypothesis, differences among
them are not due solely to differences in power against alternatives. Hence
one could consider using some of them sequentially in an attempt to narrow
down the inference on the nature of the process. For example, the Hinich
test and the White test could be used initially to find out whether the
process lacks third order nonlinear dependence and is linear in the mean. If
either test rejects its null, one could try to narrow down the nature of the
nonlinearity further by running the NEGM test to see if there is evidence of
chaos. Alternatively, if the Hinich and White tests both lead to acceptance
of the null, one could run the BDS or Kaplan test to see if the process
appears to be fully linear. If the data leads to rejection of full linearity
but acceptance of linearity in the mean, then the data may exhibit
stochastic volatility of the ARCH or GARCH\ type.

In short, the available tests provide useful information, and such
comparisons of other tests could help further to narrow down alternatives.
But ultimately we are left with the problem of isolating the nature of
detected nonlinearity or chaos to be within the structure of the economy.
This final challenge remains unsolved, especially in the case of chaos.

\subsection{Testability of Chaos within the Economy}

Recently there has been considerable criticism of the existing research on
chaos, as for example in Granger's (1994) review of Benhabib's (1992) book.
However, it is unwise to take a strong opinion (either pro or con) in that
area of research. Contrary to popular opinion within the profession, there
have been no published tests of chaos `within the structure of the economic
system', and there is very little chance that any such tests will be
available in this field for a very long time. Such tests are simply beyond
the state of the art.

All of the published tests of chaos in economic data test for evidence of
chaos in the data. If chaos is found, the test has no way of determining
whether or not the source of the chaos is from within the structure of the
economy or perhaps is from within the chaotic weather systems that surround
the planet. Considering the fact that chaos is clearly evident in many
natural phenomena, and considering the fact that natural phenomena introduce
shocks into the economy, the observation of chaotic behavior in some
economic variables should be no surprise, but should give us no reason to
believe that the economic system is chaotic, or is not chaotic.

To determine whether the source of chaos in economic data is from within the
economic system, a model of the economy must be constructed. The null
hypothesis that then must be tested is the hypothesis that the parameters of
the model are within the subset of the parameter space that supports the
chaotic bifurcation regime of the dynamic system. Currently, however, we do
not have the mathematical tools to find and characterize that subset, when
more than three parameters exist. Hence, with any usable model of any
economy, the set that defines the null hypothesis cannot be located - and no
one can test a null hypothesis that cannot be located and defined.

Since we cannot test the hypothesis, we may instead wish to consider whether
or not chaos is plausible on philosophical ground. On that basis, the
question would be whether the economy should be viewed as having evolved
naturally, as in the natural sciences, or was the product of intentional
human design by economic `engineers'. Systems intentionally designed (by
engineers) to be stable are stable and not chaotic, if designed optimally.
Nature, however, was not designed by human beings, and is chaotic - the
weather, for example, will never converge to a steady state. Which view is
more appropriate to understanding the dynamics of actual economies is not
clear.

\section{Conclusion}

We have reviewed a great deal of high quality research on nonlinear and
complex dynamics and evidence concerning chaotic nonlinear dynamics in
economic and financial time series. There are many reasons for this
interest. Chaos, for example, represents a radical change of perspective on
business cycles. Business cycles receive an endogenous explanation and are
traced back to the strong nonlinear deterministic structure that can pervade
the economic system. This is different from the (currently dominant)
exogenous approach to economic fluctuations, based on the assumption that
economic equilibria are determinate and intrinsically stable, so that in the
absence of continuing exogenous shocks the economy tends towards a steady
state, but because of stochastic shocks a stationary pattern of fluctuations
is observed.

Chaos could also help unify different approaches to structural
macroeconomics. As Grandmont (1985) has shown, for different parameter
values even the most classical of economic models can produce stable
solutions (characterizing classical economics) or more complex solutions,
such as cycles or even chaos (characterizing much of Keynesian economics).
Finally, if forecasting is a goal, the possible existence of chaos could be
exploitable and even invaluable. If, for example, chaos can be shown to
exist in asset prices, the implication would be that profitable,
nonlinearity-based trading rules exist (at least in the short run and
provided the actual generating mechanism is known). Prediction, however,
over long periods is all but impossible, due to the `sensitive dependence on
initial conditions' property of chaos.

However, as we argued in the previous section, we do not have the slightest
idea of whether or not the economy exhibits chaotic nonlinear dynamics (and
hence we are not justified in excluding the possibility). Until the
difficult problems of testing for chaos `within the structure of the
economic system' are solved, the best that we can do is to test for chaos in
economic data, without being able to isolate its source. But even that
objective has proven to be difficult. While there have been many published
tests for chaotic nonlinear dynamics, little agreement exists among
economists about the correct conclusions.

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\vspace*{-1cm}

\begin{center}
\begin{tabular}{lllll}
\multicolumn{5}{c}{TABLE 1.} \\ 
\multicolumn{5}{l}{} \\ 
\multicolumn{5}{c}{SUMMARY OF PUBLISHED RESULTS OF NONLINEARITY} \\ 
\multicolumn{5}{c}{AND CHAOS TESTING ON ECONOMIC DATA} \\ \hline\hline
\multicolumn{5}{l}{} \\ 
Study & Data & N & Tests & Results \\ 
\multicolumn{5}{l}{} \\ \hline
Barnett \& & Various weekly simple-sum & 807 & Correlation dimension & 
Evidence of nonlinearity \\ 
Chen (1988) & and Divisia monetary &  &  & and chaos in some \\ 
& aggregates &  &  & aggregates \\ 
\multicolumn{5}{l}{} \\ 
DeCoster \& & Various weekly simple-sum & 469-842 & Correlation dimension & 
Evidence of nonlinearity \\ 
Mitchell (1990) & and Divisia monetary &  &  & and chaos in some \\ 
& aggregates &  &  & aggregates \\ 
&  &  &  &  \\ 
Ramsey, Sayers, & Barnett \& Chen's (1988) & 807-5,200 & Correlation
dimension & No evidence of chaos \\ 
\& Rothman (1990) & Divisia M2 series and &  &  &  \\ 
& Scheinkman \& LeBaron's &  &  &  \\ 
& (1989) CRSP value weighted &  &  &  \\ 
& daily stock returns &  &  &  \\ 
&  &  &  &  \\ 
Serletis (1995) & Various monthly simple-sum & 396 & NEGM & Evidence that
the Divisia L \\ 
& and Divisia velocity series &  &  & velocity series is chaotic \\ 
&  &  &  &  \\ 
Brock \& & Several U.S. macroeconomic & 134 & Correlation dimension & 
Nonlinearity, but little \\ 
Sayers (1988) & time series, using (mostly) &  &  & evidence of
low-dimensional \\ 
& quarterly data from the late &  &  & chaos \\ 
& 40's to the mid 80's &  &  &  \\ 
&  &  &  &  \\ 
Frank \& & Several Canadian macroeconomic & 147 & Correlation dimension & 
Nonlinearity, but no \\ 
Stengos (1988) & time series, using (mostly) &  &  & evidence of chaos \\ 
& quarterly data since 1947 &  &  &  \\ 
&  &  &  &  \\ 
Frank, Gencay, \& & Real GNP series for Germany, & 87-108 & Correlation
dimension & Nonlinearity, but no \\ 
Stengos (1988) & Italy, Japan, and the U.K., using &  &  & evidence of chaos
\\ 
& quarterly data since 1960 &  &  &  \\ 
&  &  &  &  \\ \hline\hline
\end{tabular}

\newpage 
\end{center}

\vspace*{-1cm}

\begin{center}
\begin{tabular}{lllll}
\multicolumn{5}{c}{TABLE 2.} \\ 
\multicolumn{5}{l}{} \\ 
\multicolumn{5}{c}{SUMMARY OF PUBLISHED RESULTS OF NONLINEARITY} \\ 
\multicolumn{5}{c}{AND CHAOS TESTING ON FINANCIAL DATA} \\ \hline\hline
\multicolumn{5}{l}{} \\ 
Study & Data & N & Tests & Results \\ 
\multicolumn{5}{l}{} \\ \hline
Serletis \& & Seven East European & 438 & a. BDS & a. Not {\it iid} \\ 
Gogas (1997) & black-market exchange rates &  & b. NEGM & b. Some evidence
of chaos \\ 
&  &  & c. Gencay \& Dechert & c. No evidence of chaos \\ 
\multicolumn{5}{l}{} \\ 
Abhyankar, Copeland, & Real-time returns on four & 2,268 - 97,185 & a. BDS & 
a. Not {\it iid} \\ 
and Wong (1997) & stock-market indices &  & b. NEGM & b. No evidence of chaos
\\ 
&  &  &  &  \\ 
Abhyankar, Copeland, & FTSE 100 & 60,000 & a. Bispectral linearity test & a.
Nonlinearity \\ 
and Wong (1997) &  &  & b. BDS & b. Not {\it iid} \\ 
&  &  & c. NEGM & c. No evidence of chaos \\ 
&  &  &  &  \\ 
Hsieh (1991) & Weekly S\&P 500 and & 1,297 - 2,017 & BDS & Not {\it iid} \\ 
& CRSP value weighted returns &  &  &  \\ 
&  &  &  &  \\ 
Frank \& & Gold and silver rates & 2,900 - 3,100 & a. Correlation dimension
& a. $D_{c}=6-7$ \\ 
Stengos (1989) & of return &  & b. Kolmogorov entropy & b. Low-dimensional
chaos \\ 
&  &  &  &  \\ 
Hinich \& & Dow Jones industrial & 750 & Bispectral Gaussianity & 
Non-Gaussian and \\ 
Patterson (1989) & average &  & and linearity tests & nonlinear \\ 
&  &  &  &  \\ 
Scheinkman \& & Daily CRSP value & 5,200 & BDS & Evidence of nonlinearity \\ 
LeBaron (1989) & weighted returns &  &  &  \\ 
&  &  &  &  \\ 
Brockett, Hinich \& & 10 Common U.S. stocks & 400 & Bispectral Gaussianity & 
Non-Gaussian and \\ 
Patterson (1988) & and \$-yen spot and &  & and linearity tests & nonlinear
\\ 
& forward exchange rates &  &  &  \\ 
&  &  &  &  \\ \hline\hline
\end{tabular}
\newpage 

\begin{tabular}{lcccccc}
\multicolumn{7}{c}{TABLE 3.} \\ 
&  &  &  &  &  &  \\ 
\multicolumn{7}{c}{RESULTS OF A SINGLE-BLIND CONTROLLED COMPETITION} \\ 
\multicolumn{7}{c}{AMONG TESTS FOR NONLINEARITY AND CHAOS} \\ \hline\hline
&  &  &  &  &  &  \\ 
\multicolumn{2}{l}{} & \multicolumn{2}{c}{Small Sample} &  & 
\multicolumn{2}{c}{Large Sample} \\ \cline{3-4}\cline{6-7}
Test & Null hypothesis & Successes & Failures &  & Successes & Failures \\ 
\cline{1-6}\cline{4-7}
&  &  &  &  &  &  \\ 
Hinich & \multicolumn{1}{l}{Lack of 3rd order} & 3 & 2 &  & 3 plus ambiguous
& 1 plus ambiguous \\ 
& nonlinear dependence &  &  &  & in 1 case & in 1 case \\ 
&  &  &  &  &  &  \\ 
BDS & \multicolumn{1}{l}{Linear process} & 2 & Ambiguous &  & 5 & 0 \\ 
&  &  & in 3 cases &  &  &  \\ 
&  &  &  &  &  &  \\ 
NEGM & \multicolumn{1}{l}{Chaos} & 5 & 0 &  & 5 & 0 \\ 
&  &  &  &  &  &  \\ 
White & \multicolumn{1}{l}{Linearity in mean} & 4 & 1 &  & 4 & 1 \\ 
&  &  &  &  &  &  \\ 
Kaplan & \multicolumn{1}{l}{Linear process} & 5 & 0 &  & 5 & 0 \\ 
\hline\hline
\multicolumn{7}{l}{} \\ 
\multicolumn{7}{l}{Source: Barnett, Gallant, Hinich, Jungeilges, Kaplan, and
Jensen (1997, tables 1-4, 6-7, and 9-10).}
\end{tabular}
\newpage 
\end{center}

\end{document}
