\documentstyle[12pt]{article}
\begin{document}

\title{MARTINGALES, NONLINEARITY, AND CHAOS}
\author{WILLIAM A. BARNETT \\
%EndAName
Department of Economics, Campus Box 1208\\
Washington University\\
One Brookings Drive\\
St. Louis, Missouri 63130-4810\bigskip  \and APOSTOLOS\ SERLETIS \\
%EndAName
Department of Economics\\
The University of Calgary\\
Calgary, Alberta T2N 1N4}
\maketitle

\begin{abstract}
In this article we provide a review of the literature with respect to the
efficient markets hypothesis and chaos. In doing so, we contrast the
martingale behavior of asset prices to nonlinear chaotic dynamics, discuss
some recent techniques used in distinguishing between probabilistic and
deterministic behavior in asset prices, and report some evidence. Moreover,
we look at the controversies that have arisen about the available tests and
results, and raise the issue of whether dynamical systems theory is
practical in finance.\medskip 

{\it Keywords}: Efficient markets hypothesis, Chaotic dynamics.\medskip 

{\it JEL} classification: C22, G14\bigskip 

*We would like to thank Cars Hommes and Dee Dechert for useful comments.
\end{abstract}

\section{Introduction}

\qquad 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,
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.

Although most of the empirical tests of the efficient markets hypothesis are
based on linear models, interest in nonlinear chaotic processes has in the
recent past experienced a tremendous rate of development. There are many
reasons for this interest, one of which being the ability of such processes
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.

In this paper, we survey the recent literature with respect to the efficient
markets hypothesis and chaos. In doing so, in the next two sections we
briefly discuss the efficient markets hypothesis and some of the more recent
testing methodologies. In section 4, we provide a description of the key
features of the available tests for independence, nonlinearity, and chaos,
focusing explicit attention on each test's ability to detect chaos. In
section 5, we present a discussion of the empirical evidence on
macroeconomic and (mostly) financial data, and in section 6, we look at the
controversies that have arisen about the available tests and address some
important questions regarding the power of some of these tests. The final
section concludes.

\section{The Martingale Hypothesis}

\qquad Standard asset pricing models typically imply the `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
`fair game'{\it \ }(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 `martingale
differences'.\footnote{%
The martingale process is a special case of the more general submartingale
process. In particular, $x_{t}$ is a `submartingale' if it has the property $%
E_{t}(x_{t+1}|\Omega _{t})\geq x_{t}.$ In terms of the $(x_{t+1}-x_{t})$
process, the submartingale model implies that $E_{t}[(x_{t+1}-x_{t})|\Omega
_{t}]\geq 0$ and embodies the concept of a superfair game. LeRoy (1989, pp.
1593-4) also offers an example in which $E_{t}[(x_{t+1}-x_{t})|\Omega
_{t}]\leq 0$, in which case $x_{t}$ will be a `supermartingale', embodying
the concept of a subfair game.} 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 `efficient markets
hypothesis'.

In fact 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 `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.

In fact, Campbell, Lo, and MacKinlay (1997) distinguish between three
versions of the random walk hypothesis --- the
`independently-and-identically-distributed-returns' version, the
`independent-returns' version, and the version of `uncorrelated-returns' ---
see Campbell, Lo, and MacKinlay (1997) for more details. The martingale
difference model, by not requiring probabilistic independence between
successive price changes, 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).

\section{Tests of the Martingale Hypothesis}

\qquad 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.\footnote{%
It is to be noted that unit root tests have low power against relevent
alternatives. Also, as Granger (1995) points out, nonlinear modelling of
nonstationary variables is a new, complicated, and largely undeveloped area.
We therefore ignore this issue in this paper, keeping in mind that this is
an area for future research.}

Nelson and Plosser (1982), using the augmented Dickey-Fuller (ADF) unit root
testing procedure [see Dickey and Fuller (1981)] test the null hypothesis of
`difference-stationarity' against the `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.

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.

Finally, given that integration tests are sensitive to the class of models
considered (and may be misleading because of misspecification),
`fractionally'-integrated representations, which nest the unit-root
phenomenon in a more general model, have also been used - see 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 Poterba and Summers (1988), and Lo and 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 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
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 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''.

\section{Tests of Nonlinearity and Chaos}

\qquad Most of the empirical tests that we discussed so far 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 such reasons 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. 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. In what
follows, we turn to several univariate statistical tests for independence,
nonlinearity and chaos, that have been recently motivated by the mathematics
of deterministic nonlinear dynamical systems.

\subsection{The Correlation Dimension Test}

\qquad Grassberger and Procaccia (1983) suggested the `correlation
dimension' test for chaos. To briefly discuss this test, 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
`embedding dimension' and each $X_{t}$ is known as an `$m$-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 - in particular, from the sample size $n,$ $N=n-m+1$ $m$%
-histories can be made. 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 --- see Takens (1981).

The correlation dimension test is based on the `correlation function' (or
`correlation 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 $0$), and $\left\| .\right\| $ denotes the distance induced by the
selected norm (the `maximum norm' being the type used most often). 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.

The correlation dimension can be defined as

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

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 Barnett, Gallant, Hinich, Jungeilges, Kaplan, and
Jensen (1995, pp. 306) put it ``[i]f the only source of stochasticity is
[observational] 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.

\subsection{The BDS Test}

\qquad To deal with the problems of using the correlation dimension test,
Brock, Dechert, LeBaron, and Scheinkman (1996) devised a new statistical
test which is known as the BDS test --- see also Brock, Hsieh, and LeBaron
(1991). 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 Hinich
(1992) for a discussion of these issues.

\subsection{The Hinich Bispectrum Test}

\qquad 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 - see Hinich (1982). 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.
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}

\qquad 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 Wolf, Swift, Swinney, and 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, Nychka,
Ellner, Gallant, and 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 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 Schwartz (1978). As shown by
Gallant and 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 $\widehat{T}_{N}=\widehat{J}_{N}%
\widehat{J}_{N-1},...,\widehat{J}_{1}$.

Another very promising approach to the estimation of Lyapunov exponents
[that is similar in some respects to the Nychka et al. (1992) approach] has
also been recently proposed by Gencay and Dechert (1992). This involves
estimating all Lyapunov exponents of an unknown dynamical system. The
estimation is carried out, as in Nychka et al. (1992), by a multivariate
feedforward network estimation technique --- see Gencay and Dechert (1992)
for more details.

\subsection{The White Test}

\qquad 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 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}

\qquad Kaplan (1994) used the fact that solution paths in phase space reveal
deterministic structure that is not evident in a plot of $x_{t}$ versus $t$,
to produce 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 - see Kaplan (1994) or
Barnett et al. (1997) for more details about this procedure.

\section{Evidence on Nonlinearity and Chaos}

\qquad A number of researchers have recently focused on testing for
nonlinearity in general and chaos in particular in macroeconomic 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.\footnote{%
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)}

There is a broad consensus of support for the proposition that the
(macroeconomic) data generating processes are characterized by a pattern of
nonlinear dependence, but there is no consensus at all on whether there is
chaos in macroeconomic time series. For example, Brock and Sayers (1988),
Frank and Stengos (1988), and Frank, 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 Chen (1988),
claimed successful detection of chaos in the (demand-side) U.S. Divisia
monetary aggregates. Their conclusion was further confirmed by DeCoster and
Mitchell (1991, 1994). This published claim of successful detection of chaos
has generated considerable controversy, as in Ramsey, Sayers, and Rothman
(1990) and Ramsey and Rothman (1994), who raised questions regarding
virtually all published tests of chaos. Further results relevant to this
controversy have recently been provided by Serletis (1995).

Although the analysis of macroeconomic time series has not yet led to
particularly encouraging results (mainly due to the small samples and high
noise levels for most macroeconomic series), as can be seen from Table 1,
there is also a substantial literature testing for nonlinear dynamics on
financial data.\footnote{%
For other unpublished work on testing nonlinearity and chaos on financial
data, see Abhyankar, Copeland, and Wong (1997, Table 1).} This literature
has led to results which are as a whole more interesting and more reliable
than those of macroeconomic series, 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). As regards the main conclusions of this literature, there is
clear evidence of nonlinear dependence and some evidence of chaos.

For example, 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 Gogas (1997) test for chaos in seven East
European black market exchange rates, using the Koedijk and 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, Ghashghaie, Breymann, Peinke, Talkner, and Dodge (1996) analyze all
worldwide 1,472,241 bid-ask quotes on U.S. dollar-German mark exchange rates
between October 1, 1992 and September 30, 1993. They apply physical
principles and provide 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''.

\section{Controversies}

\qquad Clearly, 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 - see Barnett et al. (1995) on this issue.

It is also 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
email to experts in running each of the statistical tests that were entered
into the competition. The emailed 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 $2$. 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.

\section{Conclusion}

\qquad 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. The presence of dynamic noise (i.e., noise added in each
iteration step) makes it difficult and perhaps impossible to distinguish
between (noisy) high-dimensional chaos and pure randomness. The estimates of
the fractal dimension, the correlation integral, and Lyapunov exponents of
an underlying unknown dynamical system are sensitive to dynamic noise, and
the problem grows as the dimension of the chaos increases. The question of
the `impossibility' of distinguishing between high-dimensional chaos and
randomness has recently attracted some attention, as for example in
Radunskaja (1994), Bickel and B\"{u}hlmann (1996), and Takens (1997).
Analogously, Bickel and B\"{u}hlmann (1996) argue that distinguishing
between linearity and nonlinearity of a stochastic process may become
impossible as the order of the linear filter increases. In a time series
framework, it is prudent to limit such tests to the use of low order linear
filters as approximations to nonlinear processes when testing for general
nonlinearity, and tests for low dimensional chaos, when chaotic nonlinearity
is of interest --- see also Barnett et al. (1997, footnote 11).

However, in the field of economics, it is especially 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. Existing tests
cannot tell whether the source of detected chaos comes from within the
structure of the economy, or from chaotic external shocks, as from the
weather. Thus, we do not have the slightest idea of whether or not asset
prices exhibit chaotic nonlinear dynamics produced from the nonlinear
structure of the economy (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 time series 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.\newpage

\section{References}

\vskip 0.25cm \noindent Abhyankar, A. H., L. S. Copeland, and W. Wong, 1995,
Nonlinear Dynamics in Real-Time Equity Market Indices: Evidence From the UK,
Economic Journal{\it \ }105 (431), 864-880.

\vskip 0.25cm \noindent Abhyankar, A. H., L. S. Copeland, and W. Wong, 1997,
Uncovering Nonlinear Structure in Real-Time Stock-Market Indexes: The S\&P
500, the DAX, the Nikkei 225, and the FTSE-100, {\it \ }Journal of Business
and Economic Statistics{\it \ }15, 1-14.

\vskip 0.25cm \noindent Baillie, R. T., 1996, Long Memory Processes and
Fractional Integration in Econometrics, Journal of Econometrics 73, 5-59.

\vskip 0.25cm \noindent Barnett, W. A. and\ P. Chen, 1988, The
Aggregation-Theoretic Monetary Aggregates are Chaotic and Have Strange
Attractors: An Econometric Application of Mathematical Chaos, in: W. A.
Barnett, E. R. Berndt, and H. White, eds., Dynamic Econometric Modeling
(Cambridge University Press, Cambridge, U.K.).

\vskip0.25cm \noindent Barnett, W. A., A. R. Gallant, M.J. Hinich, J.
Jungeilges, D. Kaplan, and M.J. Jensen, 1997, A Single-Blind Controlled
Competition Among Tests for Nonlinearity and Chaos, Journal of Econometrics
82 (1997), 157-192.

\vskip 0.25cm \noindent Barnett, W. A., A. R. Gallant, M.J. Hinich, J.
Jungeilges, D. Kaplan, and M.J. Jensen, 1995, Robustness of Nonlinearity and
Chaos Tests to Measurement Error, Inference Method, and Sample Size, Journal
of Economic Behavior and Organization 27, 301-320.

\vskip 0.25cm \noindent Barnett, W. A., A. R. Gallant, M.J. Hinich, J.
Jungeilges, D. Kaplan, and M.J. Jensen, 1996, An Experimental Design to
Compare Tests of Nonlinearity and Chaos, in: W. A. Barnett, A. Kirman, and
M. Salmon, eds., Nonlinear Dynamics in Economics (Cambridge University
Press, Cambridge, U.K.).

\vskip 0.25cm \noindent Barnett,\ W.\ A. and M. J.\ Hinich, 1992, Empirical
Chaotic Dynamics in Economics, Annals of Operations Research 37,1-15.

\vskip0.25cm \noindent Benhabib, J., 1992, Cycles and Chaos in Economic
Equilibrium (Princeton University Press, Princeton, NJ).

\vskip0.25cm \noindent Bickel, P.J. and P. B\"{u}hlmann, 1996, What is a
Linear Process?, Proceedings of the National Academy of Science, USA,
Statistics Section 93, 12,128-12,131.

\vskip 0.25cm \noindent Brillinger, D. R., 1965, An Introduction to the
Polyspectrum, Annals of Mathematical Statistics 36, 1351-1374.

\vskip0.25cm \noindent Brock, W. A., W. D. Dechert, B. LeBaron, and J. A.
Scheinkman, 1996, A Test for Independence Based on the Correlation
Dimension, Econometric Reviews 15, 197-235.

\vskip0.25cm \noindent Brock, W. A., D. A. Hsieh, and B. LeBaron, 1991,
Nonlinear Dynamics, Chaos, and Instability: Statistical Theory and Economic
Evidence (MIT Press, Cambridge, MA).

\vskip 0.25cm \noindent Brock, W. A. and C. Sayers, 1988, Is the Business
Cycle Characterized by Deterministic Chaos?, Journal of Monetary Economics
22, 71-90.

\vskip 0.25cm \noindent Brockett, P. L., M. J. Hinich, and D. Patterson,
1988, Bispectral-Based Tests for the Detection of Gaussianity and
Nonlinearity in Time Series, Journal of the American Statistical Association
83, 657-664.

\vskip 0.25cm \noindent Campbell, J. Y., A. W. Lo, and A. C. MacKinlay,
1997, The Econometrics of Financial Markets (Princeton University Press,
Princeton, NJ).

\vskip 0.25cm \noindent Cochrane, J. H., 1988, How Big is the Random Walk in
GNP?, Journal of Political Economy 96, 893-920.

\vskip 0.25cm \noindent DeCoster, G. P. and D. W. Mitchell, 1991, Nonlinear
Monetary Dynamics, Journal of Business and Economic Statistics 9, 455-462.

\vskip 0.25cm \noindent DeCoster, G. P. and D. W. Mitchell, 1994, Reply,
Journal of Business and Economic Statistics 12, 136-137.

\vskip 0.25cm \noindent Dickey, D. A. and W. A. Fuller, 1981, Likelihood
Ratio Tests for Autoregressive Time Series With a Unit Root, Econometrica
49, 1057-1072.

\vskip 0.25cm \noindent Fama, E. F., 1970, Efficient Capital Markets: A
Review of Theory and Empirical Work, Journal of Finance 25, 383-417.

\vskip 0.25cm \noindent Fama, E. F., 1965, The Behavior of Stock Market
Prices, Journal of Business 38, 34-105.

\vskip 0.25cm \noindent Fama, E. F. and K. R. French, 1988, Permanent and
Temporary Components of Stock Prices, Journal of Political Economy 96,
246-273.

\vskip 0.25cm \noindent Frank, M., R. Gencay, and T. Stengos, 1988,
International Chaos, European Economic Review 32, 569-1584.

\vskip 0.25cm \noindent Frank, M. and T. Stengos, 1988, Some Evidence
Concerning Macroeconomic Chaos, Journal of Monetary Economics 22, 423-438.

\vskip 0.25cm \noindent Frank, M. and T. Stengos, 1989, Measuring the
Strangeness of Gold and Silver Rates of Return, Review of Economic Studies
56, 553-567.

\vskip 0.25cm \noindent Gallant, R. and H. White, 1992, On Learning the
Derivatives of an Unknown Mapping With Multilayer Feedforward Networks,
Neural Networks 5, 129-138.

\vskip 0.25cm \noindent Gencay, R. and W. D. Dechert, 1992, An Algorithm for
the $n-$Lyapunov Exponents of an $n-$Dimensional Unknown Dynamical System,
Physica D 59, 142-157.

\vskip 0.25cm \noindent Ghashghaie, S., W. Breymann, J. Peinke, P. Talkner,
and Y. Dodge, 1996, Turbulent Cascades in Foreign Exchange Markets, Nature
381, 767-770.

\vskip 0.25cm \noindent Grandmont, J.-M., 1985, On Endogenous Competitive
Business Cycles, Econometrica 53, 995-1045.

\vskip0.25cm \noindent Granger, C. W. J., 1994, Is Chaotic Economic Theory
Relevant for Economics? A Review Article of: Jess Benhabib: Cycles and Chaos
in Economic Equilibrium, Journal of International and Comparative Economics,
3, 139-145.

\vskip0.25cm \noindent Granger, C. W. J., 1995, Modelling Nonlinear
Relationships Between Extended-Memory Variables, Econometrica 63, 265-279.

\vskip 0.25cm \noindent Grassberger, P. and I. Procaccia, 1983,
Characterization of Strange Attractors, Physical Review Letters 50, 346-349.

\vskip 0.25cm \noindent Hinich, M. J., 1982, Testing for Gaussianity and
Linearity of a Stationary Time Series, Journal of Time Series Analysis 3,
169-176.

\vskip 0.25cm \noindent Hinich, M. J. and D. Patterson, 1989, Evidence of
Nonlinearity in the Trade-by-Trade Stock Market Return Generating Process,
in: W. A. Barnett, J. Geweke, and K. Shell, eds., Economic Complexity:
Chaos, Bubbles, and Nonlinearity (Cambridge University Press, Cambridge,
U.K.).

\vskip 0.25cm \noindent Hsieh, D. A., 1991, Chaos and Nonlinear Dynamics:
Applications to Financial Markets, Journal of Finance 46, 1839-1877.

\vskip 0.25cm \noindent Kaplan, D. T., 1994, Exceptional Events as Evidence
for Determinism, Physica D 73, 38-48.

\vskip 0.25cm \noindent Koedijk, K. G., and C. J. M. Kool, 1992, Tail
Estimates of East European Exchange Rates, Journal of Business and Economic
Statistics 10, 83-96.

\vskip 0.25cm \noindent Lee, T.-H., H. White, and C. W. J. Granger, 1993,
Testing for Neglected Nonlinearities in Time Series Models, Journal of
Econometrics 56, 269-290.

\vskip 0.25cm \noindent LeRoy, S. F., 1989, Efficient Capital Markets and
Martingales, Journal of Economic Literature 27, 1583-1621.

\vskip 0.25cm \noindent Lo, A. W. and A. C. MacKinlay, 1988, Stock Market
Prices Do Not Follow Random Walks: Evidence from a Simple Specification
Test, Review of Financial Studies 1, 41-66.

\vskip 0.25cm \noindent Mandelbrot, B. B., 1963, The Variation of Certain
Speculative Stock Prices, Journal of Business 36, 394-419.

\vskip 0.25cm \noindent McCaffrey, D., S. Ellner, R. Gallant, and D. Nychka,
1992, Estimating the Lyapunov Exponent of a Chaotic System with
Nonparametric Regression, Journal of the American Statistical Association
87, 682-695.

\vskip 0.25cm \noindent Nelson, C. R. and C. I. Plosser, 1982, Trends and
Random Walks in Macroeconomic Time Series, Journal of Monetary Economics 10,
139-162.

\vskip 0.25cm \noindent Nychka, D., S. Ellner, R. Gallant, and D. McCaffrey,
1992, Finding Chaos in Noisy Systems, Journal of the Royal Statistical
Society{\it \ }B 54, 399-426.

\vskip 0.25cm \noindent Perron, P., 1989, The Great Crash, the Oil Price
Shock, and the Unit Root Hypothesis, Econometrica 57, 1361-1401.

\vskip0.25cm \noindent Poterba, J. M. and L. H. Summers, 1988, Mean
Reversion in Stock Prices: Evidence and Implications, Journal of Financial
Economics 22, 27-59.

\vskip0.25cm \noindent Radunskaya, A., 1994, Comparing Random and
Deterministic Series, Economic Theory 4, 765-776.

\vskip 0.25cm \noindent Ramsey, J. B. and P. Rothman, 1994, Comment on
``Nonlinear Monetary Dynamics'' by DeCoster and Mitchell, Journal of
Business and Economic Statistics 12, 135-136.

\vskip 0.25cm \noindent Ramsey, J. B., C. L. Sayers, and P. Rothman, 1990,
The Statistical Properties of Dimension Calculations Using Small Data Sets:
Some Economic Applications, International Economic Review 31, 991-1020.

\vskip 0.25cm \noindent Ruelle, D., 1990, Deterministic Chaos: The Science
and the Fiction, Proc. R. Soc. London A 427 (1873), 241-248.

\vskip 0.25cm \noindent Scheinkman, J. A., 1990, Nonlinearities in Economic
Dynamics, Economic Journal 100, 33-48.

\vskip 0.25cm \noindent Scheinkman, J. A. and B. LeBaron, 1989, Nonlinear
Dynamics and Stock Returns,{\it \ }Journal of Business 62, 311-337.

\vskip 0.25cm \noindent Schwartz, G., 1978, Estimating the Dimension of a
Model, The Annals of Statistics{\it \ }6, 461-464.

\vskip 0.25cm \noindent Serletis, A., 1995, Random Walks, Breaking Trend
Functions, and the Chaotic Structure of the Velocity of Money, Journal of
Business and Economic Statistics 13, 453-458.

\vskip0.25cm \noindent Serletis, A. and P. Gogas, 1997, Chaos in East
European Black-Market Exchange Rates, Research in Economics 51 (1997),
359-385.

\vskip0.25cm \noindent Summers, L. H., 1986, Does the Stock Market
Rationally Reflect Fundamental Values, Journal of Finance 41, 591-601.

\vskip0.25cm \noindent Takens, F., 1981, Detecting Strange Attractors in
Turbulence, in D. Rand and L. Young, eds., Dynamical Systems and Turbulence
(Springer-Verlag, Berlin).

\vskip0.25cm \noindent Takens, F., 1997, The Effect of Small Noise on
Systems with Chaotic Dynamics, in S. J. van Strien and S. M. Verduyn Lunel,
eds., Stochastic and Spatial Structures of Dynamical Systems (North-Holland,
Amsterdam).

\vskip 0.25cm \noindent White, H., 1989, Some Asymptotic Results for
Learning in Single Hidden-Layer Feedforward Network Models, Journal of the
American Statistical Association 84, 1003-1013.

\vskip 0.25cm \noindent Wolf, A., J. B. Swift, H. L. Swinney, and J. A.
Vastano, 1985, Determining Lyapunov Exponents from a Time Series, Physica
16D 16, 285-317.

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