This document discusses the evolution of research on the Efficient Market Hypothesis (EMH) in finance. It begins by outlining the three forms of market efficiency put forth in EMH. It then describes how Mandelbrot and others challenged EMH by finding long-term dependence and non-linear relationships in asset price movements, contrary to EMH assumptions of randomness. The document outlines Mandelbrot's rescaled range statistical technique and discusses how later researchers used non-linearity tests to further analyze non-random patterns in markets. It questions the validity of conventional linear tests used to support EMH and argues considering non-linearity is crucial to understanding market efficiency.

1. National Seminar on Nonlinearity,
Complex Dynamics & Chaos in Economics
& Finance
University of
Calcutta
March 13th 2013
Nonlinearity &
Chaos in Finance:
THE Journey so far &
THE ROAD AHEAD
Dr. Vinodh Madhavan
Finance & Accounting Area
Indian Institute of Management Lucknow
Email: vinodh.madhavan@iiml.ac.in

2. Efficient Market Hypothesis (EMH)
• A major intellectual advancement in the field of Finance is the
Efficient Market Hypothesis (EMH).
• Market efficiency could be broadly classified into there versions, as
shown below.
• Weak form version of efficiency
• The past history of price movements pertaining to any
stock is already impounded in the current stock price.
• Semi-strong form of efficiency
• Asset prices should reflect all publicly available
information pertaining to the company.
• Strong-form efficiency
• Asset prices should reflect all publicly & private
information pertaining to the company.

3. Efficient Market Hypothesis (EMH)
• On the econometric front, prevalence of EMH would
mean that stock price variations are generated by a
random process, which has no long-term memory. That is,
stock price fluctuations are Independent and Identically
Distributed (IID).
• In a nutshell, market efficiency precludes possibilities to
make consistent profits via trading rules aimed at
exploiting arbitrage opportunities in the market place.

4. EMH: Acceptance followed by Dispute
• Earlier studies on Market Efficiency found little evidence of
significant autocorrelation in the short-run amidst security
prices. See Fama (1970) for a review of early literature on this
front.
• However, over time, a substantial body of literature challenging
EMH developed that touched upon aspects such as
• Positive autocorrelation of short-term returns (Lo &
MacKinlay, 1988; Conrad & Kaul, 1989)
• Predictability over the long-term horizon (DeBondt &
Thaler, 1985; Fama & French, 1988; Jegadeesh, 1991;
Poterba & Summers, 1988; Shiller, 1984; & Summers,
1986). See Fama (1991) for a survey on this front.

5. Mandelbrot and his pioneering contribution
• Mandelbrot’s pioneering work on cotton prices challenged EMH
by establishing that asset price increments indeed have a longterm memory. (Mandelbrot, 1963).
• Mandelbrot, while working with cotton prices found that the
autocorrelation of asset prices fall; but they fall more slowly than
expected; and it takes a very long time before the correlations die
out. In short, Mandelbrot found manifestations of long-term
dependence.
• Should a series exhibits long memory/long-term dependence, it
reflects persistent temporal dependence even between distant
observations (Barkoulas & Baum, 1996).

6. Inspiration behind Mandelbrot’s work
• Inspiration behind Mandelbrot’s R/S technique was the
research work associated with an Englishman named Harold
Erwin Hurst (Hurst, 1951)
• Hurst undertook path breaking studies of river Nile in 20th
century for the purpose of informing the British Government
of how high a dam should they build at Aswan, Egypt to
control for floods in extremely wet years and at the same time
create reservoirs of water for irrigation during years of
drought (Madhavan & Pruden, 2011)

7. Hurst, 1951
• According to Hurst, the size of storage reservoir R, that has to be
built by the British Government should satisfy the following
power exponent law.
l𝑜𝑔
𝑅
σ
𝑁
2
= 𝐾𝑙𝑜𝑔
𝑁
𝑅= σ
2
𝐾
Where
σ: standard deviation of cumulative sum of departures of
annual discharges from the mean annual discharge over the
years.
N is the number of years involved in the study
K is the power law exponent.

8. Mandelbrot’s R/S Technique
• Mandelbrot’s rescaled range statistic is widely used to test longterm dependence in a time series.
• Contrary to conventional statistical tests, Mandelbrot’s Classical
R/S method does not make any assumptions with regard to the
organization of the original data.
• The R/S formula simply measures whether, over varying periods
of time, the amount by which the data vary from maximum to
minimum is greater or smaller than what a researcher would
expect if each data point were independent of the prior one.
• If the outcome is different, this implies that the sequence of data
is critical.

9. Mandelbrot’s R/S Technique
• Mandelbrot’s classical R/S method requires division of the
time series into a number of subseries of varying length k.
• Then, for each value of k, R/S statistic for each subseries,
followed by the average value of R/S considering all the
subseries is calculated using the following framework
• Then, log[R(k)/S(k)] values are plotted against log k values.
• Following such a scatter plot, a least squares regression is
employed so as to fit an optimum line through different log
R/S vs. log k scatter plots.

10. Mandelbrot’s R/S Technique
• The slope of the regression line yields H, the long-range
dependence coefficient.
• In honor of Hurst and another Mathematician named Ludwig
Otto Holder, Mandelbrot termed the long-range dependence
coefficient as H.
• For a Gaussian time series, the H value should be 0.5
• An H value of 0.5 <H<1 would indicate positive long-term
dependence, while a H value of 0<H<0.5 would indicate antipersistence (otherwise called mean reversion) behavior
• A time series that exhibits long-term dependence could be
best characterized as Fractional Brownian motion (Mandelbrot
& Hudson, 2004).

11. Other Challengers
• Rogers (1997) further established that should the asset
price fluctuations be characterized by fractional Brownian
motion, this would offer a gateway to make consistent
profits via trading rules aimed at exploiting arbitrage
opportunities in the market place.
• Also, Scheinkman & Lebaron (1989) found that weekly
returns based on Centre for Research in Securities Prices
(CRSP) datasets exhibit evidence that is incompatible with
EMH. They also found evidence of nonlinearity in the
datasets.

12. Conventional EMH tests: Evidence from
Early Literature
• Most empirical tests on EMH used conventional tests such
as autocorrelation tests to explore linear predictability (or
lack-there-of) of datasets.
• Should the autocorrelations turn out to be absent,
then such asset classes were termed efficient.
• Should the autocorrelation be present, such asset
classes were termed predictable and consequently
inefficient.
• Other traditional tests that were employed to test EMH
were the runs test and unit root test.

13. Conventional EMH tests: Evidence from
Early Literature
• Unit root tests such as Augmented Dickey Fuller Test (Dickey &
Fuller, 1979, 1981) & Phillips-Perron test (Phillips & Perron,
1988) are designed to reveal whether a time series is stationary
I(0).
• Absence of stationarity/Prevalence of unit root/I(1) was
construed as evidence of market efficiency by early
researchers.
• Lo & Mackinlay (1989) subsequently demonstrated that tests such
as autocorrelation test and runs test are less powerful compared
to the variance ratio test aimed at testing for autocorrelation.
Consequently, variance ratio tests were also employed by many
researchers to test for efficiency of markets.

14. Questions about validity of conventional
EMH tests
• However, Saadi et al. (Saadi, Gandhi, & Elmawazini, 2006)
questioned the validity of many traditional tests including
variance ratio test, that were employed to test market
efficiency.
• Unit root tests are intent on findings out whether the
shocks to any asset class is temporary or permanent. Such
tests are not designed to detect predictability of asset
prices. Consequently detection of unit root cannot be
construed as a basis for support of EMH.
• Further prior studies such as Lo & MacKinlay, 1988, 1990;
Miller et al., 1994 indicate that autocorrelation amidst asset
prices could be spurious owing to thin trading.

15. Questions about validity of conventional
EMH tests
• Thin trading is all the more likely to be evidenced in small
capitalization stocks . Consequently, it takes time for new
information to get impounded in the stock price of the
small capitalization stocks.
• As a result, studies on emerging market efficiency using
conventional statistical tests are more likely to biased
owing to thin trading.

16. Role of Nonlinearity in EMH Argument
• The main criticism of Saadi et al. was that conventional
statistical tests limit themselves to exploring linear
predictability (if any) in asset movements.
• Asset return series could be linearly uncorrelated (and appear
random based on these conventional test results), but at the
same time such time series could be nonlinearly dependent.
• Until and unless nonlinearity/higher order temporal
dependence is explored appropriately, conclusive argument on
EMH would be unconvincing.

17. Nonlinearity Tests
• In an effort to capture nonlinear serial dependence, something
that was missing in prior research efforts on market efficiency,
Saadi et al., recommended BDS test (Brock, Dechert, &
Scheinkman, 1996)
• Heeding to the advice of Saadi et al., researchers subsequently
utilized the non-linearity toolkit made available by Patterson &
Ashley (2000) so as to detect both linear and non-linear
structures in financial time series.
• The toolkit contained the following tests
1.
2.
3.
4.
5.
6.
McLeod-Li test (McLeod & Li, 1983)
Engle’s Lagrange Multiplier test (Engle, 1982)
BDS Test (Brock et al., 1996)
Tsay test (Tsay, 1996)
Hinich bispectrum test (Hinich, 1982)
Hinich bicorrelation test (Hinich, 1996).

18. Nonlinearity Toolkit
• Patterson & Ashely’s nonlinearity toolkit have been used by
• Panagiotidis (2002,2005)
• Panagiotidis & Pelloni (2003)
• Ashley & Patterson (2006)
• Lim (2009)
• Lim & Brooks (2009)
• With the exception of the Hinich bispectrum test, the remaining
five tests in the non-linearity toolkit tests for serial dependence of
any kind (both linear and nonlinear).
• With regard to the bispectrum test, Ashley et al. (1986) proved that
this test is invariant to the linear filtering of the data.

19. Nonlinearity tests: Differing power
• But for the bispectrum test, the input data needs to be prewhitened so that any remaining dependence subsequently detected
by any of the remaining five non linearity tests can indicate a
nonlinear data generating mechanism.
• Results pertaining to various Monte Carlo simulations employed
by different authors indicate that not all nonlinearity tests have the
same power.
• Further, most of the nonlinearity tests have different power against
different nonlinear process and no one test dominates the other
tests (Ashely et al., 1986; Ashley & Patterson, 1989; Hsieh, 1991;
Lee, et al., 1993; Brock, et al., 1991, 1996; Barnett et al., 1997;
Patterson & Ashley, 2000)

20. No Unanimous Verdict
• As a consequence of differing power of different
nonlinearity tests, outcomes pertaining to prior studies
reflect no unanimous verdict when it comes to presence of
nonlinearity (See Lim & Brooks, 2009; Lim, 2009;
Caraiani, 2012)
• For more on the mathematics behind each of the
nonlinearity tests available, the role of outliers, and noisy
chaos, refer to Kyrtsou & Serletis, 2006; Hommes &
Manzan, 2006

21. What is Chaos?
• A chaotic process is a processes that appears to be random, but
is generated by a deterministic model. Such a process cannot be
detected using standard statistical tests such as autocorrelation
functions (Sakai & Tokumaru, 1980)
• While stochastic trends of irregular systems are explained by
exogenous shocks, chaotic systems are characterized by
fluctuations within the system (endogenous shocks), which are
caused by complex interactions amidst the system’s elements.
• Further, chaotic systems can also be defined as systems that are
characterized by Sensitive Dependence on Initial Conditions
(SDIC).

22. What is SDIC?
• Consider a time series X wherein
X(t+1) = f(x(t))
• If an infinitesimal change δx(0) is made at time t=0 (initial
condition), the at time t, a corresponding change of δx(t) will
happen.
• Now we can say that X(t) exhibits SDIC, if δx(t) grows
exponentially with t
|δx(t)| = |δx(0)|eλT
Where λ>0 is called the Lyapunov Exponent.
Source: Ruelle, 1990

23. Discovery of Chaos
• Chaos was first observed by J. Hadamard (1898) in a
special system called Geodesic flow on a manifold of
constant negative curvature.
• The philosophical importance of this discovery was later
realized by people like Duhem (1906) and Poincare
(1908)

24. Poincare, 1903
“A very small cause that escapes our notice determines a considerable effect that we
cannot fail to see, and then we say that the effect is due to chance. If we knew exactly
the laws of nature and the situation of the universe at the initial moment, we could
predict exactly the situation of that same universe at a succeeding moment.
But even if it were the case that the natural laws had no longer any secret for us, we
could still only know the initial situation approximately. If that enabled us to predict
the succeeding situation with the same approximation, that is all we require, and we
should say that the phenomenon had been predicted, that it is governed by laws.
But it is not always so; it may happen that small differences in the initial conditions
produce very great ones in the final phenomena. A small error in the former will
produce an enormous error in the latter. Prediction becomes impossible, and we have
the fortuitous phenomenon”

25. Terms pertinent to Chaos Literature
• The larger framework from which Chaos emerges from, is the so
called dynamical systems theory.
• A dynamical system consists of two parts:
• the notion of state
• A rule that describes how the state evolves (Can be visualized in
a state space)
• The coordinates of any state space system is the degrees of freedom
required to characterize a system.
• If a dynamical system’s evolution happens in continuous time, it is
termed as flow. If the same happens in discrete time, it is termed as
mapping.
• Over time, the behavior of any system would be attracted towards/
would settle down towards a smaller region of state space. Such a
region is called an attractor.

26. Terms pertinent to Chaos Literature
• Some systems do not come to rest in the long term, but they cycle
periodically in a sequence of states. For instance, pendulum clock,
human heart.
• A system can have many attractors.
• Understanding Chaos lies in understanding the simple stretching
and folding operation that takes place in a system’s state space.
• A time series characterized by long-term dependence coupled with
non-periodic cycles is termed as a fractal (Mandelbrot, 1977).
• A fractal reveals more details as it is increasingly magnified.
• For a succinct overview of historical and theoretical antecedents
behind chaos theory, refer to Crutchfield, Farmer, Packard, & Shaw,
1986.

27. GP Test (Grassberger & Procaccia, 1983)
• Grassberger & Procaccia (1983a, 1983b) developed a metric test to
identify chaotic behavior in time series data
Underlying Philosophy:
• Any unknown system that generates a time series yt is ndimensional in nature [Dimension reflects the degrees of
freedom relevant to a dynamic system]
• The input data is transformed into a series of points in an mdimensional Euclidean space by “data embedding”
• Should the input series be random, then the dimension of points
in Euclidean space (given by “correlation dimension” measure)
will increase with increase in value of m.
• On the other hand, should the underlying system that generates
yt be chaotic, then the correlation dimension will peak and will
not increase further for subsequent higher value of m.

28. Application of GP test in Finance &
Economics
• Initial analysis of financial and economic time series
offered some evidence consistent with chaos (See Barnett
& Chen, 1988; Frank & Stengos, 1989; and Sayers, 1987).
• In due course, limitations of GP test applications in the
filed of finance and economics became evident.
• Unlike natural sciences, data sets in finance and
economics are relatively small and very noisy. In such
conditions, the GP test did not work well.

29. BDS Test
• Considering the limitations of GP test when applied to small and
noisy datasets in finance and economics, an alternative test called
the BDS test (Brock, Dechert, & Scheinkman, 1987; Brock,
Dechert & Scheinkman, & LeBaron, 1996) was developed.
• Underlying Philosophy
• BDS test is actually derived from GP test
• But the null hypothesis of the test is not that time series is
chaotic, but rather that the underlying time series is
Independent & Identically Distributed (I.I.D.)
• Alternative hypothesis includes prevalence of linear, nonlinear
and/or chaotic structure

30. Application of BDS test in Finance
• Applications of BDS test in the finance arena revealed strong evidence
of nonlinear dependence but no convincing evidence of chaos (Frank
& Stengos, 1988; Hsieh, 1989, 1991; Mayfield & Mizrach, 1992;
Peters, 1991; Scheinkman & LeBaron, 1989)
• Further, Ramsey, Sayers & Rothman (1989) reevaluate prior research
findings pertaining to chaos in finance and economics, using a
procedure developed by Ramsey & Yuan (1989a, 1989b) and find
virtually no evidence of chaotic attractors of the type that were
discovered in physical sciences.
• In addition, Ruelle (1990), Eckmann & Ruelle (1992) have showed
that any proof of low dimensional chaos in short and noisy datasets is
inconclusive and suspicious. The same has been acknowledged by
LeBaron (1994)

31. Lyapunov Exponents test
• Tests such as Correlation Dimension test, and BDS test
allows for distinction between different nonlinear systems
to some extent.
• To be more specific, BDS test produces indirect evidence
of nonlinear dependence, which is necessary but not
sufficient for chaos (Barnett et al., 1995, 1997)
• A more direct test for chaos is the Lyapunov test as it
indicates the level of chaos in any underlying system as
opposed to earlier tests such as correlation dimension test
that estimate the complexity of any underlying nonlinear
system (Faggini, 2011a)

32. Calculating Lyapunov Exponent
• There are two classes of methods to
Lyapunov Exponent λ
estimate the
• Direct method proposed by Wolf, Swift, Swinney, &
Vastano (1985), wherein the Lyapunov Exponent is
based on calculation of growth rate of difference
between two trajectories with an infinitesimal
difference in their initial conditions.
• A recent regression method proposed by Nychka,
Ellner, Gallant, & McCaffrey (1992), which involves
the use of neural networks to estimate the Lyapunov
Exponent.This method is also called the NEGM test.

33. Topological Approach to Chaos
• The metric tests discussed so far, namely Correlation
Dimension, BDS test, Lyapunov Exponent are highly
sensitive to noise (Barnett & Serletis, 2000)
• As pointed earlier, datasets pertaining to economics and
finance suffer from smaller size and low signal to noise
ratio.
• To overcome this challenge, researchers devised a new
testing for chaos using topological tools (Mindlin, Hou,
Solari, Gilmore, & Tufillaro, 1990; Tufillaro, 1990;
Tufillaro, Solari, & Gilmore, 1990)

34. Topological Method: How are they different?
• Such topological methods were aimed at studying the
organization of the strange attractor (a set of points
towards which a chaotic system would converge)
• Also, the topological methods search for a more
fundamental characteristic of chaos: the tendency of a
chaotic time series to nearly, although never exactly,
repeat itself over time.
• In addition, unlike the metric approach, topological tests
preserve the time ordering of the data, and they work
very well in small and noisy datasets (Faggini, 2007,
2011a, 2011b)

35. Topological Method: How are they different?
• Finally, unlike metric tests, topological tests do not aggregate
the data. Rather topological tests such as close-returns test
would identify location of chaotic episodes within a time series.
• Two notable topological methods are
• Close-returns test (Gilmore, 1993, 1996, 2001; McKenzie,
2001)
• Recurrence Analysis (Eckmann, Kamphorst, & Ruelle,
1987)
• Recurrence Analysis involves data embedding, while Closereturns test is employed without embedding.

36. Close returns test
• Close returns test consists of two component
• A qualitative component (Close returns plot)
• This is aimed at detecting chaotic structure
• A quantitative component
• This is aimed at detecting departures from I.I.D.
• Underlying Philosophy
• Let xt be a time series whose trajectories are orbiting the
phase space. If the orbit is one period, then the trajectory
will return to the neighborhood of xt after an interval of
one.
• Therefore, if xt evolves near a periodic orbit for a sufficiently
long time, it will return to the neighborhood of xt after
some interval T.

37. Close returns test
• The criterion for closeness requires that the difference
|xT – xT+i| be smaller than the threshold value.
• So, all differences of |xT – xT+i| for T = 1 to n and i = 1 to n-1 is
computed.
• The threshold value is chosen arbitrarily
• For example: Threshold value ε may be chosen as 2% of the
largest distance between any two points |xT – xT+i|
• Then a close returns plot is constructed wherein if the distance
between any two points is lower than the threshold value chosen, then
it is coded black. If the distance happens to be larger than the
threshold value, then it is coded white.

38. Random Time Series- Close returns plot

39. Chaotic Time Series - Close returns plot

40. Construction of Histogram
• This histogram reflects the number of close returns (black dots)
for every i.This is given by
𝐻𝑖 =
Θ(ε − 𝑥 𝑇 − 𝑥 𝑇+𝑖 )
Where Θ is the Heaviside theta function
Θ(x)=1 if x≥0, and Θ(x)=0 if x<0
• For a pseudo random time series, histogram constructed
based on close returns plot would exhibit scattering around a
uniform distribution.
• For a chaotic time series such as Henon map, the histogram
would contain a series of peaks.

41. 0
i
951
901
851
801
751
701
651
601
551
501
451
401
351
301
251
201
151
101
51
1
20
39
58
77
96
115
134
153
172
191
210
229
248
267
286
305
324
343
362
381
400
419
438
457
476
495
514
533
552
571
590
609
628
647
666
685
704
723
742
761
780
799
818
837
856
875
894
913
932
951
970
989
0
1
H(i)
H(i)
Histograms: Random vs. Chaotic Time series
70
60
50
40
30
20
10
i
300
250
200
150
100
50

42. Quantification of close returns plot
• Finally, a Chi-square test aimed at detecting departures from i.i.d.
based on close-returns plot is conducted
𝑘
𝐻𝑖 − 𝐻 2
𝑖=1
2
𝜒𝑡 =
𝑛𝑝(1 − 𝑝)
Where H = n × p; n being the number of observations over which
the close-returns is counted, and
𝑇𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑐𝑙𝑜𝑠𝑒 𝑟𝑒𝑡𝑢𝑟𝑛𝑠
𝑝=
𝑇𝑜𝑡𝑎𝑙 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑝𝑙𝑜𝑡
• The calculated χ2 is then compared with the critical chi-square
value pertaining to k-1 degrees of freedom. If the ratio between χ2
and critical chi-square value is greater than 1, then i.i.d. is rejected.

43. Recurrence Plots
• Recurrence Analysis is similar to Close returns test, but it differs on
plot construction.
• Unlike close returns plot, recurrence plots are symmetric over
the main diagonal.
• Close returns plot analyses the time series directly by fixing the
threshold value ε, while Recurrence plot is based on
reconstruction of time series using data embedding and
estimation of closeness of data points, as measured by critical
phase space radius (Faggini, 2011a)
• If a time series is chaotic, then the recurrence plot would show
short segments parallel to the main diagonal.
• If a time series is random, then the recurrence plot would not
show any kind of structure.

44. Recurrence Quantification Analysis (RQA)
• At times, recurrence plots per se are not easy to interpret, because
the segments parallel to main diagonal might not be clear.
• Consequently, a quantification technique based on recurrence plots
was proposed by Zbilut & Webber (1998, 2000)
• Different measures of recurrence plots have been proposed in
literature
• Recurrence Rate (REC): Fraction of recurrence points in the
recurrence plot
• Determinism (DET): Percentage of recurrence points forming
line segments that are parallel to the main diagonal.
• Maxline(LMAX): Length of longest diagonal line, excluding the
line of identity.

45. Recurrence Quantification Analysis (RQA)
• Measures (Continued…)
• Shannon Entropy (ENT): Distribution of line segments
parallel to the main diagonal: A reflection of the complexity of
the deterministic structure.
• Laminar State (LAM): Fraction of recurrence points forming
vertical lines
• Trapping Time (TT): Average length of vertical lines: Estimate
of the mean time that a system remains at a specific state.
• For an overview of the different software packages that can be
utilized to generate recurrence plots, refer to Belaire-Franch &
Contreras (2002).

46. Three recent comprehensive papers that
have been published on Nonlinearity &
Chaos in Finance Area

47. Mishra, Sehgal, & Bhanumurthy, 2011
• This is the possibly the first systematic attempt to investigate
long-range dependence, nonlinearity and chaos in Indian stock
market.
• As part of the study, the authors considered S&P CNX Nifty,
CNX IT Index, Bank Nifty, BSE Sensex, BSE 200, and BSE 100
indices
• Study’s findings reveal strong evidence of nonlinear dependence
in daily increments of all equity indices that were analyzed.
• The authors claim that nonlinearity is multiplicative in nature
and is transmitted only through variance of the process

48. Mishra, Sehgal, & Bhanumurthy, 2011
• Conditional heteroscedasticity models were found to be adequate
to capture nonlinearity in the case of S&P CNX Nifty and BSE
Sensex only. This in turn begs the question of possibility of
deterministic chaos in the other indices considered for this study.
• Mandelbrot’s Rescaled range estimation technique was employed
by these authors and the findings reflect prevalence of long-term
dependence in all of the indices (A reflection of the failure of
random walk hypothesis)
• However, Lyapunov Exponent calculated using NEGM test
indicate prevalence of chaos in only two of the indices namely
Bank Nifty and CNX IT.

49. Bastos & Caiado, 2011
• The authors investigate the presence of deterministic dependence
in 46 international stock markets (23 developed and 23 emerging
markets) using Recurrence Quantification Analysis (RQA)
• Stock markets in countries with strong economic
interdependence were found to display similar recurrence plots.
• Butterfly shaped structure in the case of US, UK and German
Stock Markets
• Arrow shaped structure in the case of Southeast Asian
markets: Indonesia, Malaysia & Thailand
• Small distances in lower left corner of Recurrence plots for
Eastern European (Czech repuclic, Poland, & Russia) and
Latin American (Argentina, Brazil, & Chile) markets

50. Bastos & Caiado, 2011
• In terms of Determinism (DET), two largest stock markets in the
world namely US and Japan exhibited first and the third lowest
values respectively. In case of merging markets, Taiwan was found
to have lowest value of determinism (DET)
• Mean comparison (T tests) and Median Comparison (WilcoxonMann-Whitney U test) indicate difference in RQA measures
between developed and emerging markets.
• The results reiterate the notion that developed stock markets
characterized by large trading volumes and liquidity, fewer
problems of information asymmetry and opaqueness are less
predictable.

51. Bastos & Caiado, 2011
• Time-dependent RQA measures further reveal that
measures of determinism such as DET and LAM were
found to exhibit large decline during time of crises at
Indonesia and Malaysia. However they were found to be
unaffected by burst of technology bubble in 2001.

52. Barkoulas, Chakraborty, & Ouandlous, 2012
• The authors test whether the spot price of crude oil is
determined by stochastic rules or deterministic
endogenous fluctuations.
• Daily data pertaining to West Texas Intermediate (WTI)
crude oil spot prices from 1/2/1986 to 8/31/2011 was
considered for this study.
• Findings reflect absence of any chaotic component as
measured by three indications of chaos
• No stabilization of correlation dimension
• No exhibition of sensitive dependence on initial
conditions (SDIC)
• No recurrent states being exhibited in recurrence plots

53. Barkoulas, Chakraborty, & Ouandlous, 2012
• Recurrence plots of GARCH filtered oil returns suggest
that volatility clustering is a fairly adequate, but not a
complete characterization of nature of evolution of crude
oil spot market.

54. So, all things considered…
• Yes, there is a broad consensus on presence of
nonlinear dependence in financial markets.
• However, the issue is unsettled when it comes to
chaos, as there is mixed evidence in financial
markets.
• Further, the concept of chaos in financial
market happens to be highly controversial, in
the same lines as the EMH

55. Publishing on Nonlinearity and Chaos:
Some Personal Perspectives (Not
Scientifically testable propositions)

56. Thank you