This short paper is a comment on ``Testing for Nonlinear Structure and Chaos in Economic Time Series'' by Catherine Kyrtsou and Apostolos Serletis. We summarize their main results and discuss some of their conclusions concerning the role of outliers and noisy chaos. In particular, we include some new simulations to investigate whether economic time series may be characterized by low dimensional noisy chaos.

Hommes, C.H.

Manzan, S.

English

Research Papers in Economics

Testing for Nonlinear Structure and Chaos
in Economic Time Series: A Comment
Cars H. Hommes¤ and Sebastiano Manzan
CeNDEF, University of Amsterdam
August 2005, Journal of Macroeconomics, forthcoming
Abstract
This short paper is a comment on \Testing for Nonlinear Structure and Chaos in Economic
Time Series" by Catherine Kyrtsou and Apostolos Serletis. We summarize their main re-
sults and discuss some of their conclusions concerning the role of outliers and noisy chaos.
In particular, we include some new simulations to investigate whether economic time series
may be characterized by low dimensional noisy chaos.
Keywords: nonlinearity; chaos; self-organized criticality; Mackey-Glass-GARCH
JEL classi¯cation: C22; C45; C61;
¤ Corresponding author: Cars H. Hommes, CeNDEF, University of Amsterdam, Roetersstraat 11, 1018WB
Amsterdam, the Netherlands. Phone number: + 31 20 525 4246, Fax numer: + 31 20 525 4349, email:
c.h.hommes@uva.nl.
11 Introduction
Since the mid eighties several economists have tried to test for nonlinearity and in particular
for chaos in economic and ¯nancial time series (e.g. Brock and Sayers (1988) and Scheinkman
and LeBaron (1989)). In order to test for chaos, two quantities may be derived from a time
series. Firstly, one can estimate the correlation dimension measuring the fractal nature of a
possibly underlying strange attractor. Secondly, one can estimate the largest Lyapunov exponent
which, when found to be positive, measures the sensitive dependence on initial conditions so
characteristic of a chaotic system.
The methods to detect chaos however are highly sensitive to noise (see e.g. Barnett and
Serletis (2000) for an extensive discussion). In particular, estimation of the correlation dimension
turned out to be di±cult for economic and ¯nancial time series. Brock et al. (1996) developed
a statistical test for independence, known as the BDS-test, based on the correlation integral,
which can be used as a general speci¯cation test. More recently, an important step forward
has been made by Shintani and Linton (2004), who derived the asymptotic distribution of a
nonparametric neural network estimator of the Lyapunov exponent of a noisy system. Since
one frequently used de¯nition of chaos is a positive largest Lyapunov exponent, this test may be
seen as a direct test for chaos. Recently, this method has been applied by Linton and Shintani
(2003) to test for chaos in real output series from various countries. For most series they ¯nd a
statistically negative Lyapunov exponent, thus rejecting the hypothesis of chaos.
The current special issue of the Journal of Macroeconomics contains several papers on testing
for nonlinearity and chaos. In this comment we discuss the paper Univariate Tests for Non-
linear Structure, by Kyrtsou and Serletis (2005). After a discussion of the various nonlinearity
tests in the paper, we discuss the implications of their ¯ndings, in particular concerning the
question: are economic time series characterized by low-dimensional noisy chaos? This question
has generated some controversy in the last 15 years. For example, Granger (1991,1994) has
written critical reviews on modeling economic phenomena by deterministic chaotic models. In
order to shed some new light on this important issue we apply the recently developed methods
of Shintani and Linton (2004) to a simple low dimensional chaotic stock market model of Brock
and Hommes (1998) bu®eted with dynamic noise, to check the robustness of a positive estimate
2of the Lyapunov exponent.
Results
Kyrtsou and Serletis (2005) use a set of 10 tests that have power to detect nonlinearities of various
types. Nonlinearity can occur in the ¯rst moment of the process as well as in the conditional
variance (GARCH-type dynamics) or even higher moments. In addition, they consider the raw
series of daily returns of the USD/CAD exchange rate as well as a ¯ltered series where outliers
are removed. The tests suggest the following conclusions:
² Linearity in Mean: the White neural network test strongly rejects linearity for the un-
¯ltered exchange rate series, but does not reject linearity when the outliers are removed.
Inference using the Theiler surrogate data approach leads to the same conclusions. The
Bicovariance, Bispectral and Tsay statistics can also be considered as tests for linearity
of the conditional mean. They evaluate the signi¯cance of cross-products of lagged val-
ues of the time series. Another way of thinking about these tests is that they test for
linearity of the third conditional moment of the process, the skewness. These tests reject
(for both raw and ¯ltered returns) the null hypothesis of linearity, although for the Tsay
test only at the 10% signi¯cance level. Hence, it can be concluded that for the returns of
the USD/CAD exchange rate there is evidence for nonlinear dependence between the time
series and interaction terms of lagged values. Also, it can be interpreted as evidence that
the dependence occurs in the third conditional moment rather than in the ¯rst.
² Heteroskedasticity: another form of nonlinear dependence occurs when the conditional
variance is time-varying. The authors consider the McLeod-Li and Engle tests for depen-
dence in the conditional variance. The results strongly suggests the rejection of the null of
a constant second moment. This is largely in accordance with the widely accepted GARCH
e®ect in economic and ¯nancial time series. In addition, the results do not dependend on
the ¯ltering procedure for outliers.
² General Dependence Test: the BDS-test is a general test for dependence. Rejections occur
when the process has dependence in any moment of the distribution. In all cases, the
3BDS-test rejects the hypothesis of IID observations. The results are thus consistent with
the above evidence of structure in the second and third conditional moment.
² Chaos: the Lyapunov Exponent (LE) test for low-dimensional chaos clearly suggest that for
both the raw exchange rate return series as well as the ¯ltered series the LE is signi¯cantly
negative. This indicates that the series is consistent with a stochastic process rather than
a deterministic low-dimensional chaotic system. The authors note however that the results
may still be consistent with high-dimensional (noisy) chaos. In another paper by Serletis
and Shintani (2005) in this special issue similar results, i.e. a statistically signi¯cant
negative Lyapunov exponent, are found for monetary time series of Canadian and U.S.
simple-sum Divisa and currency equivalent money and velocity measures.
New Challenges
The paper of Kyrtsou and Serletis (2005) also contributes in reviving two long-standing debates
in the nonlinear economics dynamics community. The ¯rst relates to the role played by outliers
in testing for nonlinearity. The second is associated with the interpretation of the results to test
for chaos. We will now discuss these two issues in some more detail.
Outliers: exogenous or endogenous?
This issue relates to the interpretation of extreme observations: are they the results of large
exogenous shocks or are they inherently related to the dynamical behavior of the model? In
other words, are they exogenous phenomena that we better neglect in empirical work or are they
caused by strong nonlinearities? This is a very important issue, for example if we are interested in
forecasting extreme events. The exogenous view suggests that extreme events are unpredictable
and simply neglects them. The nonlinear dynamics approach views them as endogenous to the
system and is informative about their generating mechanism.
Further evidence on the relevance of the issue is provided by the authors in Section 6. They
estimate a model with a nonlinear structure in the conditional mean, the Generalized Mackey-
Glass (GMG) model (motivated by and related to the high-dimensional chaotic Mackey-Glass
system) together with a GARCH-model for the conditional variance. They found that for the
4raw returns there is strong evidence in support of the proposed model. However, when the
outliers are removed the best performing model is a simple GARCH(1,1) model.
Is the economy characterized by low-dimensional noisy chaos?
Application of the LE-test to economic time series suggests that there is no evidence to support
the positivity of the exponent and thus that we are dealing with a stochastic system. Most
experts note that the null of high dimensional chaos has not been rejected, because it is extremely
di±cult to distinguish between high dimensional chaos and randomness and one would need
extremely long time series to do so. Moreover, the tests are highly sensitive to noise and this
becomes worse when the dimension of the system increaes. But has low-dimensional noisy chaos
been rejected as a null? It is remarkable that this important question has not received much
attention in the literature. The main reason seems to be that well known chaotic maps such as
the one-dimensional quadratic logistic map and the two-dimensional quadratic H¶ enon-map, only
allow for extremely small levels of dynamic noise, because small noise easily causes the system
to diverge to in¯nity in the chaotic parameter range.
In order to shed some light on this important issue, we consider as an example the chaotic
asset pricing model with heterogeneous beliefs proposed by Brock and Hommes (1998) bu®eted
with dynamic noise. For suitable parameters in the chaotic region, we can push the dynamic
noise level to large values while keeping the system bounded, and we can thus investigate how
far we can push the noise level before the positive Lyapunov exponent of the underlying chaotic
skeleton model becomes negative due to the presence of dynamic noise. The model assumes that
agents hold di®erent beliefs about the future asset price and switch endogenously between the
di®erent beliefs types based on their past performance. The nonlinear dynamic model is
xt =
1
R
4 X
h=1
nh;t(ghxt¡1 + bh) + ¾²t (1)
nht =
e¯Uh;t¡1
P4
j=1 e¯Uj;t¡1 (2)
Uh;t¡1 = (xt¡1 ¡ Rxt¡2)(ghxt¡3 + bh ¡ Rxt¡2): (3)
Here xt denotes the deviation of price of the risky asset from its benchmark fundamental value
5(the discounted sum of expected future dividends), R > 1 is the constant gross risk free rate,
nh;t represents the discrete choice fraction of agents using belief type h, Uh;t¡1 is the pro¯t
generated by strategy h in the previous period, gh and bh characterize the linear belief with one
time lag of strategy h, and the noise term ²t is standard normally distributed with ¾ the standard
deviation of the dynamic noise component. Brock and Hommes (1998) show that for suitable
choices of the parameter values (especially when the intensity of choice ¯ to switch strategies is
high) the 4-type version of the deterministic skeleton of the model exhibits complicated, chaotic
dynamics. The stochastic version of the model (1) adds dynamic noise to the deterministic
structure. Notice that substituting eq. (2-3) into (1), the model is in fact a nonlinear di®erence
equation with three lags, i.e. it is of the form
xt = F(xt¡1;xt¡2;xt¡3) + ¾²t; (4)
which is equivalent to a 3-dimensional nonlinear ¯rst order system. An advantage of the model
is that, for suitable choices of the parameters in the chaotic region, it does not explode when
the noise interacts with the deterministic dynamics. Figure 1 shows the attractors of time series
from the deterministic case and for di®erent noise levels ¾.
Figure (1) about here
We now apply the LE-test1 to a time series (2000 observations) generated by the model
in the deterministic and stochastic case. This exercise is only for illustrative purposes and a
detailed analysis of the behavior of the LE for the stochastic system would require Monte Carlo
simulations. We used 3 lags in the estimation of the neural network (corresponding to the true
dimension 3 of the system) and 4 hidden units (corresponding to a sum of 4 sigmoid functions in
eq. (1) and similar to values used in empirical applications). The results are shown in Table 1.
For the deterministic case we ¯nd that the LE is signi¯cantly positive with an estimated value
¸ ¼ 0:135 close to the value ¸ ¼ 0:12 obtained with the direct method for estimating the LE
of Wolf et al. (1985). However, when we increase the noise level, the estimated LE becomes
smaller and even negative. Only for the smallest noise level ¾ = 0:05 we obtain a slightly,
but signi¯cantly, positive LE ¸ ¼ 0:038. For ¾ = 0:1 the estimated LE is very close to 0
6(slightly negative, but not signi¯cant). For ¾ = 0:2 we ¯nd a statistically signi¯cant negative
estimated LE ¸ ¼ ¡0:028. In terms of the inverse signal-to-noise (SN) ratios, measured as
SN = ¾=
p
var(xt), ¾ = 0:1 corresponds to SN = 0:22 and ¾ = 0:2 corresponds to SN = 0:36.
This evidence suggests that ¯nding a negative exponent does not imply that low-dimensional
noisy chaos has been rejected. In the presence of a relatively small amount of dynamic noise a
chaotic model may have a negative LE although the deterministic skeleton is chaotic.
Table (1) about here
Conclusion
Several papers in this special issue show that the evidence for nonlinearity is strong. It is not
clear which nonlinear model o®ers the best explanation for this detected structure, and this
remains an important topic for future work. Our simulations show that a fairly small amount of
dynamic noise may lead to a negative LE estimate for a noisy chaotic system. This suggests that
low-dimensional chaos may still explain a signi¯cant part of observed °uctuations in economic
and ¯nancial time series.
Notes
1We would like to thank Mototsugu Shintani for kindly providing his programs to compute the LE-statistic.
References
Barnett, W.A. and A. Serletis, 2000, Martingales, nonlinearity and chaos, Journal of
Economic Dynamics and Control 24, 703-724.
Brock, W.A., W.D. Dechert, J.A. Scheinkman and B. LeBaron, 1996, A test for indepen-
dence based on the correlation dimension, Econometric Reviews 15, 197-235.
Brock, W.A. and C.H. Hommes (1998), Heterogeneous beliefs and routes to chaos in a
simple asset pricing model, Journal of Economic Dynamics and Control 22, 1235-1274.
7Brock, W.A. and C.L. Sayers, 1988, Is the business cycle characterized by deterministic
chaos?, Journal of Monetary Economics 22, 71-90.
Granger, C.W.J., 1991, Developments in the Nonlinear Analysis of Economic Series, Scan-
dinavian Journal of Economics 93, 263-276.
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.
Kyrtsou, C. and A. Serletis, 2005, Univariate tests for nonlinear structure, Journal of
Macroeconomics, this issue.
Scheinkman, J. and B. LeBaron, 1989, Nonlinear dynamics and stock returns, Journal of
Business 62, 311-337.
Serletis, A. and M. Shintani, 2005, Chaotic monetary dynamics with con¯dence, Journal
of Macroeconomics, this issue.
Shintani, M. and O. Linton, 2003, Is there chaos in the world economy? A nonparametric
test using consistent standard errors, International Economic Review 44, 331-358.
Shintani, M. and O. Linton, 2004, Nonparametric neural network estimation of Lyapunov
exponents and a direct test for chaos, Journal of Econometrics 120, 1-33.
Wol®, A., J.B. Swift, H.L. Swinney and J.A. Vastano, 1985, Determining Lyapunov expo-
nents from a time series, Physica D 16, 285-317.
8(a) Deterministic: ¾ = 0 (b) ¾ = 0.1
(c) ¾ = 0.2 (d) ¾ = 0.4
Figure 1: Delay plots (xt¡1;xt) of the nonlinear model (1) for the deterministic case
and for 3 di®erent noise levels; (a) shows (a projection of) the strange attractor of the
deterministic skeleton. Parameters are: R = 1:01, ¯ = 90, g1 = b1 = 0, g2 = 0:9, b2 = 0:2,
g3 = 0:9, b3 = ¡0:2, g4 = 1:01 and b4 = 0.
9SN LE - (3,4)
Deterministic 0 0.135
(13.6)
¾ = 0.05 0.12 0.038
(3.53)
¾ = 0:1 0.22 -0.003
(-0.313)
¾ = 0:2 0.36 -0.028
(-2.24)
¾ = 0:3 0.48 -0.057
(-5.44)
¾ = 0:4 0.55 -0.07
(-5.79)
Table 1: LE estimates (with t-statistics in parenthesis) of the
neural network model with 3 lags and 4 hidden units for time series
of 2000 observations for various noise levels ¾. SN is the (inverse)
Signal-to-Noise ratio de¯ned as ¾=
p
var(xt).
10

A test for independence based on the correlation dimension,

Chaotic monetary dynamics with con¯dence,

Determining Lyapunov exponents from a time series,

Developments in the Nonlinear Analysis of Economic Series,

Heterogeneous beliefs and routes to chaos in a simple asset pricing model,

Is chaotic economic theory relevant for economics? A review article of: Jess Benhabib: Cycles and chaos in economic equilibrium,

Is the business cycle characterized by deterministic chaos?,

Is there chaos in the world economy? A nonparametric test using consistent standard errors,

Martingales, nonlinearity and chaos,

Nonlinear dynamics and stock returns,

Nonparametric neural network estimation of Lyapunov exponents and a direct test for chaos,

Univariate tests for nonlinear structure,

Testing for Nonlinear Structure and Chaos in Economic Time Series: A Comment

http://www1.fee.uva.nl/cendef/publications/papers/HomMan.pdf

Testing for Nonlinear Structure and Chaos in Economic Time Series: A Comment

Abstract

Similar works

Full text

Available Versions

Research Papers in Economics