Nonlinearity & chaos

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Nonlinearity & Chaos in Finance: THE Journey so far & THE ROAD AHEAD Dr. Vinodh Madhavan Finance & Accounting Area Indian Institute of Management Lucknow Email: [email protected] National Seminar on Nonlinearity, Complex Dynamics & Chaos in Economics & Finance University of Calcutta March 13 th 2013

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Transcript of Nonlinearity & chaos

Page 1: Nonlinearity  & chaos

Nonlinearity & Chaos in Finance:

THE Journey so far & THE ROAD AHEAD

Dr. Vinodh Madhavan

Finance & Accounting Area

Indian Institute of Management Lucknow

Email: [email protected]

National Seminar on Nonlinearity,

Complex Dynamics & Chaos in Economics

& Finance

University of

Calcutta

March 13th 2013

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Efficient Market Hypothesis (EMH)

• A major intellectual advancement in the field of Finance is theEfficient Market Hypothesis (EMH).

• Market efficiency could be broadly classified into there versions, asshown below.

• Weak form version of efficiency

• The past history of price movements pertaining to anystock is already impounded in the current stock price.

• Semi-strong form of efficiency

• Asset prices should reflect all publicly availableinformation pertaining to the company.

• Strong-form efficiency

• Asset prices should reflect all publicly & privateinformation pertaining to the company.

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Efficient Market Hypothesis (EMH)

• On the econometric front, prevalence of EMH wouldmean that stock price variations are generated by arandom process, which has no long-term memory. That is,stock price fluctuations are Independent and IdenticallyDistributed (IID).

• In a nutshell, market efficiency precludes possibilities tomake consistent profits via trading rules aimed atexploiting arbitrage opportunities in the market place.

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EMH: Acceptance followed by Dispute

• Earlier studies on Market Efficiency found little evidence ofsignificant autocorrelation in the short-run amidst securityprices. See Fama (1970) for a review of early literature on thisfront.

• However, over time, a substantial body of literature challengingEMH 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.

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Mandelbrot and his pioneering contribution

• Mandelbrot’s pioneering work on cotton prices challenged EMHby establishing that asset price increments indeed have a long-term memory. (Mandelbrot, 1963).

• Mandelbrot, while working with cotton prices found that theautocorrelation of asset prices fall; but they fall more slowly thanexpected; and it takes a very long time before the correlations dieout. In short, Mandelbrot found manifestations of long-termdependence.

• Should a series exhibits long memory/long-term dependence, itreflects persistent temporal dependence even between distantobservations (Barkoulas & Baum, 1996).

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Inspiration behind Mandelbrot’s work

• Inspiration behind Mandelbrot’s R/S technique was theresearch work associated with an Englishman named HaroldErwin Hurst (Hurst, 1951)

• Hurst undertook path breaking studies of river Nile in 20th

century for the purpose of informing the British Governmentof how high a dam should they build at Aswan, Egypt tocontrol for floods in extremely wet years and at the same timecreate reservoirs of water for irrigation during years ofdrought (Madhavan & Pruden, 2011)

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Hurst, 1951

• According to Hurst, the size of storage reservoir R, that has to bebuilt by the British Government should satisfy the followingpower exponent law.

l𝑜𝑔𝑅

σ= 𝐾𝑙𝑜𝑔

𝑁

2

𝑅 = σ𝑁

2

𝐾

Where

σ: standard deviation of cumulative sum of departures ofannual discharges from the mean annual discharge over theyears.

N is the number of years involved in the study

K is the power law exponent.

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• Mandelbrot’s rescaled range statistic is widely used to test long-term dependence in a time series.

• Contrary to conventional statistical tests, Mandelbrot’s ClassicalR/S method does not make any assumptions with regard to theorganization of the original data.

• The R/S formula simply measures whether, over varying periodsof time, the amount by which the data vary from maximum tominimum is greater or smaller than what a researcher wouldexpect if each data point were independent of the prior one.

• If the outcome is different, this implies that the sequence of datais critical.

Mandelbrot’s R/S Technique

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Mandelbrot’s R/S Technique

• Mandelbrot’s classical R/S method requires division of thetime 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 thesubseries 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 isemployed so as to fit an optimum line through different logR/S vs. log k scatter plots.

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Mandelbrot’s R/S Technique

• The slope of the regression line yields H, the long-rangedependence coefficient.

• In honor of Hurst and another Mathematician named LudwigOtto Holder, Mandelbrot termed the long-range dependencecoefficient 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-termdependence, while a H value of 0<H<0.5 would indicate anti-persistence (otherwise called mean reversion) behavior

• A time series that exhibits long-term dependence could bebest characterized as Fractional Brownian motion (Mandelbrot& Hudson, 2004).

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Other Challengers

• Rogers (1997) further established that should the assetprice fluctuations be characterized by fractional Brownianmotion, this would offer a gateway to make consistentprofits via trading rules aimed at exploiting arbitrageopportunities in the market place.

• Also, Scheinkman & Lebaron (1989) found that weeklyreturns based on Centre for Research in Securities Prices(CRSP) datasets exhibit evidence that is incompatible withEMH. They also found evidence of nonlinearity in thedatasets.

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Conventional EMH tests: Evidence from Early Literature

• Most empirical tests on EMH used conventional tests suchas autocorrelation tests to explore linear predictability (orlack-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 assetclasses were termed predictable and consequentlyinefficient.

• Other traditional tests that were employed to test EMHwere the runs test and unit root test.

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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 stationaryI(0).

• Absence of stationarity/Prevalence of unit root/I(1) wasconstrued as evidence of market efficiency by earlyresearchers.

• Lo & Mackinlay (1989) subsequently demonstrated that tests suchas autocorrelation test and runs test are less powerful comparedto the variance ratio test aimed at testing for autocorrelation.Consequently, variance ratio tests were also employed by manyresearchers to test for efficiency of markets.

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Questions about validity of conventional EMH tests

• However, Saadi et al. (Saadi, Gandhi, & Elmawazini, 2006)questioned the validity of many traditional tests includingvariance ratio test, that were employed to test marketefficiency.

• Unit root tests are intent on findings out whether theshocks to any asset class is temporary or permanent. Suchtests are not designed to detect predictability of assetprices. Consequently detection of unit root cannot beconstrued as a basis for support of EMH.

• Further prior studies such as Lo & MacKinlay, 1988, 1990;Miller et al., 1994 indicate that autocorrelation amidst assetprices could be spurious owing to thin trading.

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Questions about validity of conventional EMH tests

• Thin trading is all the more likely to be evidenced in smallcapitalization stocks . Consequently, it takes time for newinformation to get impounded in the stock price of thesmall capitalization stocks.

• As a result, studies on emerging market efficiency usingconventional statistical tests are more likely to biasedowing to thin trading.

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Role of Nonlinearity in EMH Argument

• The main criticism of Saadi et al. was that conventionalstatistical tests limit themselves to exploring linearpredictability (if any) in asset movements.

• Asset return series could be linearly uncorrelated (and appearrandom based on these conventional test results), but at thesame time such time series could be nonlinearly dependent.

• Until and unless nonlinearity/higher order temporaldependence is explored appropriately, conclusive argument onEMH would be unconvincing.

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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 subsequentlyutilized the non-linearity toolkit made available by Patterson &Ashley (2000) so as to detect both linear and non-linearstructures in financial time series.

• The toolkit contained the following tests

1. McLeod-Li test (McLeod & Li, 1983)

2. Engle’s Lagrange Multiplier test (Engle, 1982)

3. BDSTest (Brock et al., 1996)

4. Tsay test (Tsay, 1996)

5. Hinich bispectrum test (Hinich, 1982)

6. Hinich bicorrelation test (Hinich, 1996).

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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 remainingfive tests in the non-linearity toolkit tests for serial dependence ofany kind (both linear and nonlinear).

• With regard to the bispectrum test, Ashley et al. (1986) proved thatthis test is invariant to the linear filtering of the data.

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Nonlinearity tests: Differing power

• But for the bispectrum test, the input data needs to be pre-whitened so that any remaining dependence subsequently detectedby any of the remaining five non linearity tests can indicate anonlinear data generating mechanism.

• Results pertaining to various Monte Carlo simulations employedby different authors indicate that not all nonlinearity tests have thesame power.

• Further, most of the nonlinearity tests have different power againstdifferent nonlinear process and no one test dominates the othertests (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)

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No Unanimous Verdict

• As a consequence of differing power of differentnonlinearity tests, outcomes pertaining to prior studiesreflect no unanimous verdict when it comes to presence ofnonlinearity (See Lim & Brooks, 2009; Lim, 2009;Caraiani, 2012)

• For more on the mathematics behind each of thenonlinearity tests available, the role of outliers, and noisychaos, refer to Kyrtsou & Serletis, 2006; Hommes &Manzan, 2006

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What is Chaos?

• A chaotic process is a processes that appears to be random, butis generated by a deterministic model. Such a process cannot bedetected using standard statistical tests such as autocorrelationfunctions (Sakai &Tokumaru, 1980)

• While stochastic trends of irregular systems are explained byexogenous shocks, chaotic systems are characterized byfluctuations within the system (endogenous shocks), which arecaused by complex interactions amidst the system’s elements.

• Further, chaotic systems can also be defined as systems that arecharacterized by Sensitive Dependence on Initial Conditions(SDIC).

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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 (initialcondition), the at time t, a corresponding change of δx(t) willhappen.

• Now we can say that X(t) exhibits SDIC, if δx(t) growsexponentially with t

|δx(t)| = |δx(0)|eλT

Where λ>0 is called the Lyapunov Exponent.

Source: Ruelle, 1990

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Discovery of Chaos

• Chaos was first observed by J. Hadamard (1898) in aspecial system called Geodesic flow on a manifold ofconstant negative curvature.

• The philosophical importance of this discovery was laterrealized by people like Duhem (1906) and Poincare(1908)

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Poincare, 1903

“A very small cause that escapes our notice determines a considerable effect that wecannot fail to see, and then we say that the effect is due to chance. If we knew exactlythe laws of nature and the situation of the universe at the initial moment, we couldpredict 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, wecould still only know the initial situation approximately. If that enabled us to predictthe succeeding situation with the same approximation, that is all we require, and weshould 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 conditionsproduce very great ones in the final phenomena. A small error in the former willproduce an enormous error in the latter. Prediction becomes impossible, and we havethe fortuitous phenomenon”

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Terms pertinent to Chaos Literature

• The larger framework from which Chaos emerges from, is the socalled 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 ina state space)

• The coordinates of any state space system is the degrees of freedomrequired to characterize a system.

• If a dynamical system’s evolution happens in continuous time, it istermed as flow. If the same happens in discrete time, it is termed asmapping.

• Over time, the behavior of any system would be attracted towards/would settle down towards a smaller region of state space. Such aregion is called an attractor.

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Terms pertinent to Chaos Literature

• Some systems do not come to rest in the long term, but they cycleperiodically in a sequence of states. For instance, pendulum clock,human heart.

• A system can have many attractors.

• Understanding Chaos lies in understanding the simple stretchingand folding operation that takes place in a system’s state space.

• A time series characterized by long-term dependence coupled withnon-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 antecedentsbehind chaos theory, refer to Crutchfield, Farmer, Packard, & Shaw,1986.

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GP Test (Grassberger & Procaccia, 1983)

• Grassberger & Procaccia (1983a, 1983b) developed a metric test toidentify chaotic behavior in time series data

Underlying Philosophy:

• Any unknown system that generates a time series yt is n-dimensional in nature [Dimension reflects the degrees offreedom relevant to a dynamic system]

• The input data is transformed into a series of points in an m-dimensional Euclidean space by “data embedding”

• Should the input series be random, then the dimension of pointsin Euclidean space (given by “correlation dimension” measure)will increase with increase in value of m.

• On the other hand, should the underlying system that generatesyt be chaotic, then the correlation dimension will peak and willnot increase further for subsequent higher value of m.

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Application of GP test in Finance & Economics

• Initial analysis of financial and economic time seriesoffered some evidence consistent with chaos (See Barnett& Chen, 1988; Frank & Stengos, 1989; and Sayers, 1987).

• In due course, limitations of GP test applications in thefiled of finance and economics became evident.

• Unlike natural sciences, data sets in finance andeconomics are relatively small and very noisy. In suchconditions, the GP test did not work well.

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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

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Application of BDS test in Finance

• Applications of BDS test in the finance arena revealed strong evidenceof 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 researchfindings pertaining to chaos in finance and economics, using aprocedure developed by Ramsey & Yuan (1989a, 1989b) and findvirtually no evidence of chaotic attractors of the type that werediscovered in physical sciences.

• In addition, Ruelle (1990), Eckmann & Ruelle (1992) have showedthat any proof of low dimensional chaos in short and noisy datasets isinconclusive and suspicious. The same has been acknowledged byLeBaron (1994)

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Lyapunov Exponents test

• Tests such as Correlation Dimension test, and BDS testallows for distinction between different nonlinear systemsto some extent.

• To be more specific, BDS test produces indirect evidenceof nonlinear dependence, which is necessary but notsufficient for chaos (Barnett et al., 1995, 1997)

• A more direct test for chaos is the Lyapunov test as itindicates the level of chaos in any underlying system asopposed to earlier tests such as correlation dimension testthat estimate the complexity of any underlying nonlinearsystem (Faggini, 2011a)

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Calculating Lyapunov Exponent

• There are two classes of methods to estimate theLyapunov Exponent λ

• Direct method proposed by Wolf, Swift, Swinney, &Vastano (1985), wherein the Lyapunov Exponent isbased on calculation of growth rate of differencebetween two trajectories with an infinitesimaldifference in their initial conditions.

• A recent regression method proposed by Nychka,Ellner, Gallant, & McCaffrey (1992), which involvesthe use of neural networks to estimate the LyapunovExponent.This method is also called the NEGM test.

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Topological Approach to Chaos

• The metric tests discussed so far, namely CorrelationDimension, BDS test, Lyapunov Exponent are highlysensitive to noise (Barnett & Serletis, 2000)

• As pointed earlier, datasets pertaining to economics andfinance suffer from smaller size and low signal to noiseratio.

• To overcome this challenge, researchers devised a newtesting for chaos using topological tools (Mindlin, Hou,Solari, Gilmore, & Tufillaro, 1990; Tufillaro, 1990;Tufillaro, Solari, & Gilmore, 1990)

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Topological Method: How are they different?

• Such topological methods were aimed at studying theorganization of the strange attractor (a set of pointstowards which a chaotic system would converge)

• Also, the topological methods search for a morefundamental characteristic of chaos: the tendency of achaotic time series to nearly, although never exactly,repeat itself over time.

• In addition, unlike the metric approach, topological testspreserve the time ordering of the data, and they workvery well in small and noisy datasets (Faggini, 2007,2011a, 2011b)

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• Finally, unlike metric tests, topological tests do not aggregatethe data. Rather topological tests such as close-returns testwould 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 Close-returns test is employed without embedding.

Topological Method: How are they different?

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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 trajectorywill return to the neighborhood of xt after an interval ofone.

• Therefore, if xt evolves near a periodic orbit for a sufficientlylong time, it will return to the neighborhood of xt aftersome intervalT.

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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 iscomputed.

• The threshold value is chosen arbitrarily

• For example: Threshold value ε may be chosen as 2% of thelargest distance between any two points |xT – xT+i|

• Then a close returns plot is constructed wherein if the distancebetween any two points is lower than the threshold value chosen, thenit is coded black. If the distance happens to be larger than thethreshold value, then it is coded white.

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Random Time Series- Close returns plot

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Chaotic Time Series - Close returns plot

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Construction of Histogram

• This histogram reflects the number of close returns (black dots)for every i.This is given by

𝐻𝑖 = Θ(ε − 𝑥𝑇 − 𝑥𝑇+𝑖 )

WhereΘ 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 constructedbased on close returns plot would exhibit scattering around auniform distribution.

• For a chaotic time series such as Henon map, the histogramwould contain a series of peaks.

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Histograms: Random vs. Chaotic Time series

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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

𝜒𝑡2 = 𝑖=1𝑘 𝐻𝑖 −𝐻

2

𝑛𝑝(1 − 𝑝)

Where H = n × p; n being the number of observations over whichthe close-returns is counted, and

𝑝 =𝑇𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑐𝑙𝑜𝑠𝑒 𝑟𝑒𝑡𝑢𝑟𝑛𝑠

𝑇𝑜𝑡𝑎𝑙 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑝𝑙𝑜𝑡

• The calculated χ2 is then compared with the critical chi-squarevalue 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.

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Recurrence Plots

• Recurrence Analysis is similar to Close returns test, but it differs onplot construction.• Unlike close returns plot, recurrence plots are symmetric over

the main diagonal.

• Close returns plot analyses the time series directly by fixing thethreshold value ε, while Recurrence plot is based onreconstruction of time series using data embedding andestimation of closeness of data points, as measured by criticalphase space radius (Faggini, 2011a)

• If a time series is chaotic, then the recurrence plot would showshort segments parallel to the main diagonal.

• If a time series is random, then the recurrence plot would notshow any kind of structure.

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Recurrence Quantification Analysis (RQA)

• At times, recurrence plots per se are not easy to interpret, becausethe segments parallel to main diagonal might not be clear.

• Consequently, a quantification technique based on recurrence plotswas proposed by Zbilut &Webber (1998, 2000)

• Different measures of recurrence plots have been proposed inliterature

• Recurrence Rate (REC): Fraction of recurrence points in therecurrence plot

• Determinism (DET): Percentage of recurrence points formingline segments that are parallel to the main diagonal.

• Maxline(LMAX): Length of longest diagonal line, excluding theline of identity.

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• Measures (Continued…)

• Shannon Entropy (ENT): Distribution of line segmentsparallel to the main diagonal: A reflection of the complexity ofthe deterministic structure.

• Laminar State (LAM): Fraction of recurrence points formingvertical lines

• Trapping Time (TT): Average length of vertical lines: Estimateof the mean time that a system remains at a specific state.

• For an overview of the different software packages that can beutilized to generate recurrence plots, refer to Belaire-Franch &Contreras (2002).

Recurrence Quantification Analysis (RQA)

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Three recent comprehensive papers that have been published on Nonlinearity &

Chaos in Finance Area

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Mishra, Sehgal, & Bhanumurthy, 2011

• This is the possibly the first systematic attempt to investigatelong-range dependence, nonlinearity and chaos in Indian stockmarket.

• As part of the study, the authors considered S&P CNX Nifty,CNX IT Index, Bank Nifty, BSE Sensex, BSE 200, and BSE 100indices

• Study’s findings reveal strong evidence of nonlinear dependencein daily increments of all equity indices that were analyzed.

• The authors claim that nonlinearity is multiplicative in natureand is transmitted only through variance of the process

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• Conditional heteroscedasticity models were found to be adequateto capture nonlinearity in the case of S&P CNX Nifty and BSESensex only. This in turn begs the question of possibility ofdeterministic chaos in the other indices considered for this study.

• Mandelbrot’s Rescaled range estimation technique was employedby these authors and the findings reflect prevalence of long-termdependence in all of the indices (A reflection of the failure ofrandom walk hypothesis)

• However, Lyapunov Exponent calculated using NEGM testindicate prevalence of chaos in only two of the indices namelyBank Nifty and CNX IT.

Mishra, Sehgal, & Bhanumurthy, 2011

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Bastos & Caiado, 2011

• The authors investigate the presence of deterministic dependencein 46 international stock markets (23 developed and 23 emergingmarkets) using Recurrence Quantification Analysis (RQA)

• Stock markets in countries with strong economicinterdependence were found to display similar recurrence plots.

• Butterfly shaped structure in the case of US, UK and GermanStock Markets

• Arrow shaped structure in the case of Southeast Asianmarkets: Indonesia, Malaysia &Thailand

• Small distances in lower left corner of Recurrence plots forEastern European (Czech repuclic, Poland, & Russia) andLatin American (Argentina, Brazil, & Chile) markets

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• In terms of Determinism (DET), two largest stock markets in theworld namely US and Japan exhibited first and the third lowestvalues respectively. In case of merging markets, Taiwan was foundto have lowest value of determinism (DET)

• Mean comparison (T tests) and Median Comparison (Wilcoxon-Mann-Whitney U test) indicate difference in RQA measuresbetween developed and emerging markets.

• The results reiterate the notion that developed stock marketscharacterized by large trading volumes and liquidity, fewerproblems of information asymmetry and opaqueness are lesspredictable.

Bastos & Caiado, 2011

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• Time-dependent RQA measures further reveal thatmeasures of determinism such as DET and LAM werefound to exhibit large decline during time of crises atIndonesia and Malaysia. However they were found to beunaffected by burst of technology bubble in 2001.

Bastos & Caiado, 2011

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Barkoulas, Chakraborty, & Ouandlous, 2012

• The authors test whether the spot price of crude oil isdetermined by stochastic rules or deterministicendogenous fluctuations.

• Daily data pertaining to West Texas Intermediate (WTI)crude oil spot prices from 1/2/1986 to 8/31/2011 wasconsidered for this study.

• Findings reflect absence of any chaotic component asmeasured by three indications of chaos

• No stabilization of correlation dimension

• No exhibition of sensitive dependence on initialconditions (SDIC)

• No recurrent states being exhibited in recurrence plots

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• Recurrence plots of GARCH filtered oil returns suggestthat volatility clustering is a fairly adequate, but not acomplete characterization of nature of evolution of crudeoil spot market.

Barkoulas, Chakraborty, & Ouandlous, 2012

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So, all things considered…

• Yes, there is a broad consensus on presence ofnonlinear dependence in financial markets.

• However, the issue is unsettled when it comes tochaos, as there is mixed evidence in financialmarkets.

• Further, the concept of chaos in financialmarket happens to be highly controversial, inthe same lines as the EMH

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Publishing on Nonlinearity and Chaos: Some Personal Perspectives (Not

Scientifically testable propositions)

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Thank you