"This paper investigates the dynamics in a simple present discounted value asset pricing model with heterogeneous beliefs. Agents choose from a Þnite set of predictors of future prices of a risky asset and revise their ÔbeliefsÕ in each period in a boundedly rational way, according to a ÔÞtness measureÕ such as past realized proÞts. Price ßuctuations are thus driven by an evolutionary dynamics between di¤erent expectation schemes (Ôrational animal spiritsÕ). Using a mixture of local bifurcation theory and numerical methods, we investigate possible bifurcation routes to complicated asset price dynamics. In particular, we present numerical evidence of strange, chaotic attractors when the intensity of choice to switch prediction strategies is high."
model
"This paper develops a model of the social and economic interaction of speculators in a securities or foreign exchange market. Both chartist and fundamentalist strategies are pursued by traders. The formalization of chartists behavior combines elements of mimetic contagion and trend chasing leading to waves of optimism or pessimism. Furthermore, changes of strategies from chartist to fundamentalist behavior and vice versa occur because speculators compare the performance of both strategies. The dynamic system under study encompasses the time development of the distribution of attitudes among traders as well as price adjustment. Chaotic attractors are found within a broad range of parameter values. The distributions of returns derived from chaotic trajectories of the model share important characteristics of empirical data: they exhibit high peaks around the mean as well as fat tails (leptokurtosis) and become less leptokurtotic under time aggregation."
model
"This paper presents stylized facts concerning the spot intra-daily foreign exchange markets. It first describes intra-daily data and proposes a set of definitions for the variables of interest. Empirical regularities of the foreign exchange intra-daily data are then grouped under three major topics: the distribution of price changes, the process of price formation and the heterogeneous structure of the market. The stylized facts surveyed in this paper shed new light on the market structure that appears composed of heterogeneous agents. It also poses several challenges such as the definition of price and of the time-scale, the concepts of risk and efficiency, the modeling of the markets and the learning process."
PETERS, E.E., 1991. Chaos and Order in the Capital Markets. [Cited by 232] (15.95/year)
found chaos
\citeasnoun{Peters91} found chaos in equity markets.
[BEST PAPER!]
"...there is no evidence of low complexity chaotic behavior in stock returns"
no chaos
"We have found strong evidence to reject the hypothesis that stock returns are IID. The cause does not appear to be either regime changes or chaotic dynamics. Rather, the cause appear to be conditional heteroskedasticity (e.g. predictable variance changes)."
Hsieh, 1990
In an excellent paper, \citeasnoun{Hsieh91} found no evidence of low complexity chaotic behavior in stock returns.
ALEXANDER, C., 2001. Market models: a guide to financial data analysis. Wiley. [Cited by 79] (14.55/year)
Taken together, it should be clear that a large amount of cleaned and pre-whitened high-frequency data is necessary in order to apply these algorithms. However, not all the research that has been published for and against chaos in capital markets has been based on empirical methods that meet these stringent criteria. Therefore it is not surprising that opinions are very mixed. Those that claim to have found chaos in the financial markets include Peters (1991), Medio (1992) and de Grauwe et al. (1993). But many more papers find no conclusive evidence of chaos, among them Scheinkman and LeBaron (1989), Hsieh (1991), Tata and Vassilicos (1991), Liu et al. (1992), Alexander and Giblin (1994), Drunat et al. (1996), and Abhyankar et al. (1997).
Alexander (2001), page 403
excellent, but not original
FARMER, J.D., 1999. Physicists Attempt TO Scale THE Ivory Towers OF Finance. Arxiv preprint adap-org/9912002. [Cited by 78] (11.92/year)
"Another reason for skepticism about power laws in economics is that sloppy statistical analysis has led to mistakes in the past. In the 1980s, there was considerable interest in the possibility that price changes might be described by a lowdimensional chaotic attractor. Physics and biology have many examples where the existence of low-dimensional chaos is unambiguous. Why not economics? Based on a numerical computation of fractal dimension, several researchers claimed to observe low-dimensional chaos in price series. Such computations are done by measuring the coarse-grained size of a set, in this case a possible attractor of returns in a state space whose variables are lagged returns, as a function of the scale of the coarse-graining. If this behaves as a power law in the limit where the scale is small, it implies low-dimensional chaos. But it is very easy to be fooled when performing such calculations. It is critical to test against a carefully formulated null hypothesis.15 More careful statistical analysis by José Scheinkman and Blake LeBaron showed that the claims of low-dimensional chaos in price series were not well-justified.16 While nonlinearity is clearly present, there is no convincing evidence of low-dimensionality. The power-law scaling that people thought they saw was apparently just an artifact of the finite size of their data sets."
Farmer (1999)
BOOK (which I have)
\citeasnoun{BrockHsiehLeBaron91} conclude that the evidence for the presence of deterministic low-dimensional chaotic generators in economic and financial data is not very strong.
"Simple deterministic systems are capable of generating chaotic output that "mimics" the output of stochastic systems. For this reason, algorithms have been developed to distinguish between these two alternatives. These algorithms and related statistical tests are also useful in detecting the presence of nonlinear dependence in time series. In this article, the authors apply these procedures to stock returns and find evidence that indicates the presence of nonlinear dependence on weekly returns from the Center for Research in Security Prices value-weighted index."
Carol Alexander says this paper says no chaos?
review paper says not tested
Clyde and Osler paper says "consistent with chaos"
A single-blind controlled competition among tests for nonlinearity and chaos
Journal of Econometrics, Volume 82, Issue 1, 1997, Pages 157-192
William A. Barnett, A. Ronald Gallant, Melvin J. Hinich, Jochen A. Jungeilges, Daniel T. Kaplan and Mark J. Jensen
Implications of Nonlinear Dynamics for Financial Risk Management
David A. Hsieh
The Journal of Financial and Quantitative Analysis, Vol. 28, No. 1. (Mar., 1993), pp. 41-64.
not chaos
BROCK, W.A. and C.L. SAYERS, 1986. Is the business cycle characterized by deterministic chaos? Journal of Monetary Economics 22, 71-90. [Cited by 90] (4.60/year)
Tests are conducted on U.S. macroeconomic data for the presence of low-dimensional deterministic chaos, where ‘deterministic chaos’ is defined as in Grandmont's (1985) Walras-Bowley lecture or in Brock (1986). The idea is to test for evidence of endogenous instability causing the business cycle. Low-order autoregressions are hard to reject with our methods. Evidence of chaos is weak but our tests may be too weak to detect it. Evidence of nonlinearity is present in employment 1950-I to 1983-IV, unemployment 1949-I to 1982-IV, monthly post-war industrial production, and pig iron production 1877-1937.
abstract: "This article tests for nonlinear dependence and chaos in real-time returns on the world's four most important stock-market indexes. Both the Brock-Dechert-Scheinkman and the Lee. White, and Granger neural-network-based tests indicate persistent nonlinear structure in the series. Estimates of the Lyapunov exponents using the Nychka, Ellner, Gallant, and McCaffrey neural-net method and the Zeng, Pielke, and Eyckholt nearest-neighbor algorithm confirm the presence of nonlinear dependence in the returns on all indexes but provide no evidence of low-dimensional chaotic processes. Given the sensitivity of the results to the estimation parameters, we conclude that the data are dominated by a stochastic component."
\citeasnoun{AbhyankarCopelandWong97} tested the world's four most important stock-market indexes: the S&P 500, the DAX, the Nikkei 225 and the FTSE-100 and found no evidence of low-dimensional chaotic processes.
BARNETT, William A. and Apostolos SERLETIS, 2000. Martingales, nonlinearity, and chaos. Journal of Economic Dynamics and Control, Volume 24, Issues 5-7, June 2000, Pages 703-724. [Cited by 25] (4.51/year)
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."
[GOOD REVIEW PAPER]
"We consider two ways of distinguishing deterministic time-series from stochastic white noise; the Grassberger-Procaccia correlation exponent test and the Brock, Dechert, Scheinkman (or BDS) test. Using simulated data to test the power of these tests, the correlation exponent test can distinguish white noise from chaos. It cannot distinguish white noise from chaos mixed with a small amount of white noise. With i.i.d. as the null, the BDS correctly rejects the null when the data are deterministic chaos. Although the BDS test may also reject the null even when the data are stochastic, it may be useful in distinguishing between linear and nonlinear stochastic processes."
theoretical
Abhyankar, A.H., Copeland, L.S., Wong, W., 1995. Nonlinear dynamics in real-time equity market
indices: evidence from the UK. Economic Journal 105 (431), 864}880.
YES
\citeasnoun{AbhyankarCopelandWong95} tested for the presence of chaos in the FTSE 100 Index using a six month sample of about 60,000 minute-by-minute returns and found little to support the view that the process is chaotic at any frequency.
"Interest has been growing in testing for nonlinearity and chaos in economic data, but much controversy has arisen about the available results. This paper explores the reasons for these empirical difficulties. We apply five tests for nonlinearity or chaos to various monetary aggregate data series. We find that the inferences vary across tests for the same data, and within tests for varying sample sizes and various methods of aggregation of the data. Robustness of inferences in this area of research seems to be low and may account for the controversies surrounding empirical claims of nonlinearity and chaos in economics."
economics, rather than finance
BENHABIB, J., 1992. Cycles and chaos in economic equilibrium. Princeton, NJ: Princeton University Press. [Cited by 36] (2.66/year)
[book]"In recent years economists have begun to use the techniques of non-linear dynamics to show that some apparently erratic and turbulent economic phenomena reflect subtle underlying patterns. How do cyclic and chaotic dynamics arise in economic models of equilibrium? How can empirical methods be used to detect nonlinearities and cyclic and chaotic structures in economic models? In examining these questions, this book brings together the most significant work that has been done to date in economics-based chaos theory. Selected here particularly for the economist who is not a specialist in chaos theory, the essays, some previously unpublished and others not widely available, describe a new tool for understanding business cycles, stabilization policy, and forecasting."
economics, rather than markets
"Imitative and contrarian behaviours are the two typical opposite attitudes of investors in stock markets. We introduce a simple model to investigate their interplay in a stock market where agents can take only two states, bullish or bearish. Each bullish (bearish) agent polls m ‘friends’ and changes her opinion to bearish (bullish) if (i) at least m?hb (m?bh) among the m agents inspected are bearish (bullish) or (ii) at least m?hh>m?hb (m?bb>m?bh) among the m agents inspected are bullish (bearish). The condition (i) ((ii)) corresponds to imitative (antagonistic) behaviour. In the limit where the number N of agents is infinite, the dynamics of the fraction of bullish agents is deterministic and exhibits chaotic behaviour in a significant domain of the parameter space {?hb,?bh,?hh,?bb,m}. A typical chaotic trajectory is characterized by intermittent phases of chaos, quasi-periodic behaviour and super-exponentially growing bubbles followed by crashes. A typical bubble starts initially by growing at an exponential rate and then crosses over to a nonlinear power-law growth rate leading to a finite-time singularity. The reinjection mechanism provided by the contrarian behaviour introduces a finite-size effect, rounding off these singularities and leads to chaos. We document the main stylized facts of this model in the symmetric and asymmetric cases. This model is one of the rare agent-based models that give rise to interesting non-periodic complex dynamics in the ‘thermodynamic’ limit (of an infinite number N of agents). We also discuss the case of a finite number of agents, which introduces an endogenous source of noise superimposed on the chaotic dynamics."
model
YANG, S.R. and B.W. BRORSEN, 1993. Nonlinear Dynamics of Daily Futures Prices: Conditional Heteroskedasticity or Chaos?. Journal of Futures Markets. [Cited by 35] (2.61/year)
Seung-Ryong Yang and B. Wade Brorsen
Nonlinear dynamics of daily futures prices: Conditional heteroskedasticity or chaos?
Journal of Futures Markets
Volume 13, Issue 2, Date: April 1993, Pages: 175-191
\citeasnoun{YangBrorsen93} found evidence of nonlinearity in several futures markets, which was consistent with deterministic chaos in about half of the cases.
"Yang and Brorsen (1993) found evidence of nonlinearity in several futures markets, which is consistent with deterministic chaos in about half of the cases."
"The primary hypothesis of this article is that technical modeling methods may represent crude but useful ways of exploring nonlinear qualities in data. More specifically, it is proposed that graphical technical analysis may be restated in terms of attractors and strange attractors. Further, it is suggested that technical analysis methods may allow prediction on systems of higher dimension than nonlinear methods do at this time. An objective algorithm that identifies technical patterns is applied to highdimension nonlinear data, and provides support for the primary hypothesis. This article is distinguished from previous studies in that (a) a specific linkage/equivalence between technical analysis and nonlinear forecasting is proposed, and (b) statistically significant evidence in support of this specific linkage/equivalence is presented.
If this link is valid, it has important implications for the future study of technical and nonlinear analysis, which should be studied jointly, and also suggests that other disciplines applying nonlinear analysis might benefit from applying technical methods."
theoretical
"This paper investigates the existence of a deterministic nonlinear structure in the stock returns of the Athens Stock Exchange (Greece), an emerging capital market. The analysis utilizes the concepts of correlation dimension and Kolmogorov entropy, and it also includes a forecasting experiment. Application of the BDS statistical test to raw and filtered returns series suggests the presence of nonlinearities. The findings provide very weak, at best, evidence in support of a nonlinear deterministic data generating process."
\citeasnoun{BarkoulasTravlos98} investigated the existence of a deterministic nonlinear structure in the stock returns of the Athens Stock Exchange (an emerging capital market) and found no strong evidence of chaos.
BARNETT, W.A. and P. CHEN, 1988. The Aggregation-Theoretic Monetary Aggregates are Chaotic and Have Strange Attractors: An …. Dynamic Econometric Modeling. [Cited by 41] (2.34/year)
Barnett WA., Chen P. The aggregation theoretic monetary aggregates are chaotic and have strange attractors: an econometric application of mathematical chaos. In: Barnett WA, Berndt ER, White H (Eds.), Dynamic econometric modeling. Cambridge University Press: Cambridge, UK; 1988. p.199-246.
economics, not finance
Abstract: "The predictability of rates of return on gold and silver are examined. Econometric tests do not reject the martingale hypothesis for either asset. This failure to reject is shown to be misleading. Correlation dimension estimates indicate a structure not captured by ARCH. The correlation dimension is between 6 and 7 while the Kolmogorov entropy is about 0.2 for both assets. The evidence is consistent with a nonlinear deterministic data generating process underlying the rates of return. The evidence is certainly not sufficient to rule out the possibility of some degree of randomness being present."
\citeasnoun{FrankStengos89} examined gold and silver returns and found that the correlation dimension is between 6 and 7 while the Kolmogorov entropy is about 0.2 for both assets.
Abstract: "The random-walk (white-noise) model and the harmonic model are two polar models in linear systems. A model in between is color chaos, which generates irregular oscillations with a narrow frequency (color) band. Time-frequency analysis is introduced for evolutionary time-series analysis. The deterministic component from noisy data can be recovered by a time-variant filter in Gabor space. The characteristic frequency is calculated from the Wigner decomposed distribution series. It is found that about 70 percent of fluctuations in Standard & Poor stock price indexes, such as the FSPCOM and FSDXP monthly series, detrended by the Hodrick-Prescott (HP) filter, can be explained by deterministic color chaos. The characteristic period of persistent cycles is around three to four years. Their correlation dimension is about 2.5. The existence of persistent chaotic cycles reveals a new perspective of market resilience and new sources of economic uncertainties. The nonlinear pattern in the stock market may not be wiped out by market competition under nonequilibrium situations with trend evolution and frequency shifts. The color-chaos model of stock-market movements may establish a potential link between business-cycle theory and asset-pricing theory."
chaos in business cycles
Abstract: Both academic and applied researchers studying financial markets and other economic series have become interested in the topic of chaotic dynamics. The possibility of chaos in financial markets opens important questions for both economic theorists as well as financial market participants. This paper will clarify the empirical evidence for chaos in financial markets and macroeconomic series emphasizing what exactly is known about these time series in terms of forecastability and chaos. We also compare these two concepts from a financial market perspective contrasting the objectives of the practitioner with those of the economic researchers. Finally, we will speculate on the impact of chaos and nonlinear modelling on future economic research.
review paper
DECOSTER, G.P., W.C. LABYS and D.W. MITCHELL, 1992. Evidence of chaos in commodity futures prices. Journal of Futures Markets. [Cited by 29] (2.01/year)
Gregory P. Decoster, Walter C. Labys, Douglas W. Mitchell
Evidence of chaos in commodity futures prices
Journal of Futures Markets
Volume 12, Issue 3, Date: June 1992, Pages: 291-305
"In a study representative of most chaos research, DeCoster, Labys and Mitchell (1993) search for evidence of chaos in commodity futures prices. The authors apply standard tests, as described below, and conclude that the null hypothesis of non linear structure cannot be rejected."
they did provide correlation dimension based evidence of the presence of nonlinear structure.
silver, copper, sugar, and coffee futures by DeCoster, Labys, and Mitchell (1992).
\citeasnoun{DecosterLabysMitchell92} searched for evidence of chaos in commodity futures (silver, copper, sugar and coffee) prices and found evidence of nonlinear structure.
Abstract: "We test for the presence of low-dimensional chaotic structure in crude oil, heating oil, and unleaded gasoline futures prices from the early 1980s. Evidence on chaos will have important implications for regulators and short-term trading strategies. While we find strong evidence of non-linear dependencies, the evidence is not consistent with chaos. Our test results indicate that ARCH-type processes, with controls for seasonal variation in prices, generally explain the non-linearities in the data. We also demonstrate that employing seasonally adjusted price series contributes to obtaining robust results via the existing tests for chaotic structure. Maximum likelihood methodologies, that are robust to the non-linear dynamics, lend support for Samuelson’s hypothesis on contract-maturity effects in futures price-changes. However, the tests for chaos are not found to be sensitive to the maturity effects in the futures contracts. The results are robust to controls for the oil shocks of 1986 and 1991."
\citeasnoun{Adrangi-etal01} tested for the presence of low-dimensional chaotic structure in crude oil, heating oil and unleaded gasoline futures prices from the early 1980s and found no evidence of chaos.
Abstract: "This article presents the results of tests for nonlinear dependence in the daily prices of the Standard & Poor's Index and the National Association of Securities Dealers Automated Quotations System 100 Stock Index. Deterministic chaos is rejected by two of three recently developed empirical tests. The methodology for implementing these diagnostic tests for financial time series is discussed, along with a route for future research to follow."
\citeasnoun{Willey92} tested the daily prices of the Standard \& Poor's Index and the National Association of Securities Dealers Automated Quotations System 100 Stock Index. Deterministic chaos was rejected by two of three recently developed empirical tests.
On Determining the Dimension of Real-Time Stock-Price Data
E. Scott Mayfield; Bruce Mizrach
Journal of Business & Economic Statistics, Vol. 10, No. 3. (Jul., 1992), pp. 367-374.
Abstract
We estimate the dimension of high-frequency stock-price data using the correlation integral of Grassberger and Procaccia. The data, even after filtering, appear to be of low dimension. To control for dependence in higher moments, we use a new technique known as the method of delays in our reconstruction. Delaying the data leads dimension estimates similar to random processes. We conclude that the data are either of low dimension with high entropy or nonlinear but of high dimension.
\cite{MayfieldMizrach92} estimate the dimension of the S\&P 500 stock index (sampled at approximately 20-second intervals) and conclude that the data are either of low dimension with high entropy or nonlinear but of high dimension.
Chaos in East European black market exchange rates
Research in Economics, Volume 51, Number 4, December 1997, pp. 359-385.
Apostolos Serletis and Periklis Gogas
In this paper we test for deterministic chaos in seven East European black market exchange rates, using Koedijk and Kool's (1992, \textit{Journal of Business and Economic Statistics}, \emph{10}, 83--96) monthly data from January 1955 through May 1990. In doing so we use three (nonparametric) inference methods, the BDS (Brock \textit{et al.}, 1996, \textit{Econometric Reviews}, \emph{15}, 197--235) test for whiteness, the Lyapunov exponent estimator of Nychka \textit{et al.} (1992, \textit{Journal of the Royal Statistical Society}, \emph{12}, 135--136) as well as the Lyapunov exponent estimator of Gencay and Dechert (1992, \textit{Physica D}, \emph{59}, 142--157). We find some consistency in inference across methods, and we conclude, based on the Nychka \textit{et al.} (1992) estimator, that there is evidence consistent with a chaotic non-linear generation process in only two out of seven series.
\citeasnoun{SerletisGogas97} test for deterministic chaos in seven East European black market exchange rates and conclude that there is evidence consistent with a chaotic non-linear generation process in only two out of seven series.
Abstract: "The purpose of this article is to characterize linear and nonlinear serial dependence in daily futures price changes. The daily prices of four futures are included in this study: (i) S&P 500; (ii) Japanese yen; (iii) Deutsche mark; and (iv) Eurodollar. Our major empirical findings are: (i) Based on the results of nonlinearity tests (that is, the BDS, the Q2, and the TAR-F tests), we found all futures price changes contain nonlinearity in the series; (ii) a GARCH model can explain the source of nonlinearity for three out of four series; (iii) a threshold autoregressive model and autoregressive volatility model can adequately represent nonlinear dynamics of S&P 500 series; and (iv) deterministic chaos is not evident in the scaled residuals from the nonlinear time series models. Hence we favor a statistical time series approach to represent the data-generating mechanism of futures price changes."
\citeasnoun{GaoWang99} examine the daily prices of four futures contracts (S\&P 500, Japanese yen, Deutsche mark and Eurodollar) and find no evidence of deterministic chaos.
CHORAFAS, D.N. and R.L. TRIPPI, 1994. Chaos Theory in the Financial Markets. Probus. [Cited by 15] (1.30/year)
bad BOOK
ANDREOU, A.S., G. PAVLIDES and A. KARYTINOS, 2000. Nonlinear Time-Series Analysis of the Greek Exchange-Rate Market, International Journal of Bifurcation and Chaos, Vol. 10, No. 7 (2000) 1729-1758. [Cited by 7] (1.26/year)
Using concepts from the theory of chaos and nonlinear dynamical systems, a time-series analysis is performed on four major currencies against the Greek Drachma. The R/S analysis provided evidence for fractality due to noisy chaos in only two of the data series, while the BDS test showed that all four systems exhibit nonlinearity. Correlation dimension and related tests, as well as Lyapunov exponents, gave consistent results, which did not rule out the possibility of deterministic chaos for the two possibly fractal series, rejecting though the occurrence of a simple low-dimensional attractor, while the other two series seemed to have followed a behavior close to that of a random signal. SVD analysis, used to filter away noise, strongly supported the above findings and provided reliable evidence for the existence of an underlying system with a limited number of degrees-of-freedom only for those series found to exhibit fractality, while it revealed a noise domination over the remaining two. These results were further confirmed through a forecasting attempt using artificial neural networks.
\citeasnoun{AndreouPavlidesKarytinos00} examined four major currencies against the Greek Drachma and found evidence of chaos in two out of four.
The time evolution of prices and savings in a stock market is modeled by a discrete time nonlinear dynamical system. The model proposed has a unique and unstable steady-state, so that the time evolution is determined by the nonlinear effects acting out of the equilibrium. The nonlinearities strongly influence the kind of long-run dynamics of the system. In particular, the global geometric properties of the noninvertible map of the plane, whose iteration gives the evolution of the system, are important to understand the global bifurcations which change the qualitative properties of the asymptotic dynamics. Such global bifurcations are studied by geometric and numerical methods based on the theory of critical curves, a powerful tool for the characterization of the global dynamical properties of noninvertible mappings of the plane. The model unfolds more complex chaotic and unpredictable trajectories as a consequence of increasing agents' ``speculative'' or ``capital gain realizing'' attitudes. The global analysis indicates that, for some ranges of the parameter values, the system has several coexisting attractors, and it may not be robust with respect to exogenous shocks due to the complexity of the basins of attraction.
model
Weekly changes for the period 1980 to 1994 in six major stock indices (the US, Korea, Taiwan, Japan, Singapore and Hong Kong) and the World Index are examined. Also examined are the corresponding foreign exchange rates between the US and these five countries. Using spectral analysis, techniques of nonlinear dynamics and ordinary least squares regression, evidence of varying levels of market integration are documented. Some of the time series examined exhibit nonlinear dependencies.
\citeasnoun{Sewell-etal96} examined weekly changes for the period 1980 to 1994 in six major stock indices (the US, Korea, Taiwan, Japan, Singapore and Hong Kong) and the World Index as well as the corresponding foreign exchange rates between the US and the other five countries. They concluded that "[t]hese results do not prove the existence of chaos in these markets but are consistent with its existence in some cases."
Abstract: "The analysis of products’ price behaviour continues to be an important empirical issue. This study contributes to the current literature on price dynamics of products by examining for the presence of chaos and non-linear dynamics in daily oil products for the Rotterdam and Mediterranean petroleum markets. Previous studies using only one invariant, such as the correlation dimension may not effectively determine the chaotic structure of the underlying time series. To obtain better information on the time series structure, a framework is developed, where both invariant and non-invariant quantities were also examined. In this paper various invariants for detecting a chaotic time series were analysed along with the associated Brock’s theorem and Eckman–Ruelle condition, to return series for the prices of oil products. An additional non-invariant quantity, the BDS statistic, was also examined. The correlation dimension, entropies and Lyapunov exponents show strong evidence of chaos in a number of oil products considered."
\citeasnoun{PanasNinni00} found strong evidence of chaos in daily oil products for the Rotterdam and Mediterranean petroleum markets.
Bispectral-Based Tests for the Detection of Gaussianity and Linearity in Time Series
Patrick L. Brockett; Melvin J. Hinich; Douglas Patterson
Journal of the American Statistical Association, Vol. 83, No. 403. (Sep., 1988), pp. 657-664.
Abstract: "Statistical techniques have been developed that use estimated bispectrum values to test whether a sample of a time series is consistent with the hypothesis that the observations are generated by a linear process. The magnitude of the test statistics indicates the amount of divergence between the observations and the linear model hypothesis. It is important to investigate such a divergence, since the usual linear model coefficients can be shown to be biased in the face of nonlinear time series structure. The tests presented here can thus be considered diagnostic as well as confirmatory. These tests are applied to a variety of real series previously modeled with linear models. The results indicate nonlinear models may yield better results, because many of the series analyzed appear to have considerable nonlinear lagged interactions."
linearity tests
Empirical and theoretical investigations of chaotic phenomena in
macroeconomic systems are presented. Basic issues and techniques in testing
economic aggregate movements are discussed. Evidence of low dimensional
strange attractors is found in several empirical monetary aggregates. A
continuous time deterministic model with delayed feedback is proposed to
describe the monetary growth. Phase transition from periodic to chaotic motion
occurs in the model. The model offers an explanation of the multiperiodicity and
irregularity in business cycles and of the low-dimensionality of chaotic monetary
attractors. Implications in monetary control policy and a new approach to
forecasting business cycles are suggested.
economics, not finance
Frank M, Gencay R, Stengos T (1988a) International chaos, European Economic Review 32: 1569-1584
macro economic
Abstract: "Tests are conducted on quarterly macroeconomic data for Italy, Japan, the United Kingdom and West Germany to check for the presence of deterministic chaos. Deterministic chaos is rejected using a number of fairly recently developed empirical tools. In the process we found surprisingly strong evidence of nonlinearities in the case of Japan. The evidence indicates that Japan is also the most stable of the countries studied. A satisfactory modelling of Japanese data will need to allow for suitable nonlinearity."
macroeconomic
FRANK, M. and T. STENGOS, 1988. Some Evidence Concerning Macroeconomic Chaos. Journal of Monetary Economics. [Cited by 21] (1.14/year)
Frank M, Stengos T (1988b) Some evidence concerning macroeconomic chaos, J. of Monetary Economics 22:
423-438
macroeconomic
SAVIT, R., 1988. When random is not random: An introduction to chaos in market prices. Journal of Futures Markets, Volume 8, Issue 3, Date: June 1988, Pages: 271-290. [Cited by 21] (1.14/year)
Robert Savit
When random is not random: An introduction to chaos in market prices
Journal of Futures Markets
Volume 8, Issue 3, Date: June 1988, Pages: 271-290
BLANK, S.C., 1990. " Chaos" in futures markets?: a nonlinear dynamical analysis. [Cited by 17] (1.09/year)
Steven C. Blank
``Chaos'' in futures markets? A nonlinear dynamical analysis
Journal of Futures Markets
Volume 11, Issue 6, Date: December 1991, Pages: 711-728
"Several studies, such as Blank (1991), and Decoster, Labys and Mitchell (1992), haveoffered evidencethatfutures pricesappearto follow low dimensional chaotic dynamics."
\citeasnoun{Blank91} offered evidence that futures prices appear to follow low dimensional chaotic dynamics.
"Price series that are 21.5 years long for six agricultural futures markets, corn, soybeans, wheat, hogs, coffee and sugar, possess characteristics consistent with nonlinear dynamics. Three nonlinear models, ARCH, long memory and chaos, are able to produce these symptoms. Using daily, weekly and monthly data for the six markets, each of these models is tested against the martingale difference null, one-by-one. Standard ARCH tests suggest that all series might contain ARCH effects, but further diagnostics show that the series are not ARCH processes, failing to reject the null. A long-memory technique, the AFIMA model, fails to find long-memory structures in the data, except for sugar. This allows chaos analysis to be applied directly to the raw data. Carefully specifying phase space, and utilizing correlation dimension and Lyapunov exponent together, the remaining five price series are found to be chaotic processes."
\citeasnoun{WeiLeuthold98} looked at six agricultural futures markets---corn, soybeans, wheat, hogs, coffee and sugar--and found that five of them (all except sugar) were chaotic processes.
TRIPPI, R.R., 1995. Chaos & nonlinear dynamics in the financial markets: theory, evidence, and applications. Chicago: Irwin Professional Pub. [Cited by 11] (1.04/year)
BOOK with separate authors
CHORAFAS, D.N., 1994. Chaos theory in the financial markets: applying fractals, fuzzy logic, genetic algorithms, swarm …. Chicago, Ill.: Probus Pub. Co. [Cited by 11] (0.95/year)
HINICH, M.J. and D. PATTERSON, 1989. Evidence of nonlinearity in the trade-by-trade stock market return generating process. … complexity: chaos, sunspots, bubbles and nonlinearity- …. [Cited by 16] (0.92/year)
Hinich, M.J., Patterson, D., 1989. Evidence of nonlinearity in the trade-by-trade stock market return
generating process. In: Barnett, W.A., Geweke, J., Shell, K. (Eds.), Economic Complexity: Chaos,
Bubbles, and Nonlinearity. Cambridge University Press, Cambridge, UK.
yes?
BROCK, W.A. and W.D. DECHERT, 1990. Nonlinear dynamical systems: instability and chaos in economics. [Cited by 12] (0.77/year)
"The purpose of this paper is: (i) set out enough of the mathematics of dynamical systems from the perspective of chaos theory so that the endogenous emergence of chaotic dynamics and other highly complex dynamics may be understood; (ii) explain measures of instability and complexity; (iii) explain how one tests for the presence of chaos and other complex nonlinear dynamics in time series data; and (iv) explain how these concepts have been applied in economics."
URBACH, R.M.A., 2000. Footprints of Chaos in the Markets: Analyzing Non-Linear Time Series in Financial Markets and Other …. London; New York: Financial Times Prentice Hall. [Cited by 4] (0.72/year)
BOOK
PAPAIOANNOU, G. and A. KARYTINOS, 1995. Nonlinear time series analysis of the stock exchange: The case of an emerging market. International Journal of Bifurcation and Chaos. [Cited by 7] (0.66/year)
The extent to which daily return data from the Athens' Stock Exchange Index exhibits non-linear and chaotic characteristics is investigated by employing multiple tests from both the Economics' and the Natural Sciences' fields. The BDS test and Rescale Range (R/S) analysis provide evidence for non-linearity and fractality due to noisy chaos, respectively, while methods of chaotic dynamics, like correlation dimension and related tests, as well as, Lyapunov exponents, give consistent results, which do not rule out the possibility of deterministic chaos. The occurrence of a simple low-dimensional attractor, is not supported. However, noise filtered data by the use of SVD analysis and FIR filters, gives reliable evidence for the existence of an underlying dynamical system with a limited number of degrees-of-freedom.
[quite good]
BLANK, S.C., 1991. Chaos. futures markets. [Cited by 9] (0.62/year)
GRANGER, C.W.J., 1994. Is Chaotic Economic Theory Relevant for Economics? A Review Article of: Jess Benhabib: Cycles and …. Journal of International and Comparative Economics. [Cited by 7] (0.61/year)
VASSILICOS, J.C., A. DEMOS and F. TATA, 1993. No Evidence of Chaos But Some Evidence of Multifractals in the Foreign Exchange and the Stock …. Crilly, AJ, Earnshaw, RA, Jones, H.(Hrsg.)(1993): …. [Cited by 7] (0.56/year)
BARNETT, W.A., et al., 1996. An experimental design to compare tests of nonlinearity and chaos. Nonlinear dynamics in economics (Cambridge University Press, …. [Cited by 5] (0.52/year)
LEMOS, M. and H.H. STOKES, 1998. A Single-Blind Controlled Competition Among Tests for Nonlinearity and Chaos; Further Results. [Cited by 3] (0.40/year)
HIBBERT, D.B. and I.F. WILKINSON, 1994. Chaos in the dynamics of markets. Journal of the Academy of Marketing Science. [Cited by 4] (0.32/year)
BOLDRIN, M., et al., 1989). Economic Complexity: Chaos, Sunspots, Bubbles and Nonlinearity. Cambridge University Press, Cambridge,. [Cited by 4] (0.24/year)
GUILLAUME, D., 1996. Chaos, randomness and order in the foreign exchange markets. Essays on the modelling of the markets. Leuven, KU Leuven, …. [Cited by 2] (0.21/year)
SERLETIS, A. and P. DORMAAR, 1996. Chaos and nonlinear dynamics in futures markets. Nonlinear Dynamics in Economics. [Cited by 1] (0.10/year)
FRIDSON, M.S., 1994. Chaos theory in the financial markets: Applying fractals. Financial Analysts Journal. [Cited by 1] (0.09/year)
TATA, T. and C. VASSILACOS, 1991. Is there chaos in economic time series? A study of the stock and the foreign exchange markets. Financial Market Group Discussion Paper. [Cited by 1] (0.07/year)
no chaos
TVEDE, L., 1992. What ‘Chaos' Really Means in Financial Markets. Futures. [Cited by 1] (0.07/year)
BARNETT, William A., Alfredo MEDIO and Apostolos SERLETIS, 1997. Nonlinear and Complex Dynamics in Economics [Cited by 2]
"...we do not have the slightest idea of whether or not the economy exhibits chaotic nonlinear dynamics (and hence we are not justified in excluding the possibility)."
Barnett, Medio and Serletis, 1997
EDGAR, P., 1995. … for the S&P 500 in Chaos & Nonlinear Dynamics in the Financial Markets, Robert Trippi (editor). [not cited]
EDGAR, P., lip;. Chaos and Order in the Capital Markets. A New View of Cycles, Prices and Market Volatility (John &h. [Cited by 1]
EDGAR, P.E., 991). Chaos and Order in the Capital Markets, A New View of Cycles, Prices and Market Volatility. John Wiley Sons, Inc., New York. [Cited by 2]
ELDRIDGE, M., B. CHRISTOPHER and M. IRENE, 1995. Evidence of Chaos in the S&P Chash Index in Chaos & Nonlinear Dynamics in the Financial Markets, …. [not cited] (0/year)
GILMORE, C.G., 1992. Financial markets and the theory of chaos. [not cited]
GUILLAUME, D., 1995. … Attractor in the Foreign Exchange Markets? In in Chaos & Nonlinear Dynamics in the Financial Markets …. [not cited]
HIEMSTRA, Y. and R.R. TRIPPI, 995). Chaos & Nonlinear Dynamics in the Financial Markets: Theory, Evidence and Applications. Irwin, Chicago, IL,. [Cited by 1]
MURRAY, F. and S. THANASIS, 1995. … Silver Rates of Return in Chaos & Nonlinear Dynamics in the Financial Markets, Robert Trippi (editor …. [not cited]
SHIRREFF, D., ), S. Efficient markets and the quants' descent into chaos. Euromoney, o. Jg.(19. [Cited by 1] (?/year)
SPIGHT, J.L., A comparison of autoregressive and chaos theory models in the forecasting of commodities markets. [not cited]
TENORIO, Manoel F., Carlos E. PEDREIRA and Nitzi M. ROEHL, The Cotton Time Series: A Study of the Competition Series Behavior and Statistics, Nonlinear Financial Forecasting - Proceedings of the First INFFC, edited by Randall B. Caldwell
"We find no evidence of determinism (although, of course, these algorithms can only detect low-dimensional chaos)."
Wilkin and Vinson
J. Drunat, G. Dufr?enot, C. Dunis and L. Mathieu
Stochastic or chaotic dynamics in high frequency exchange rates?' in C. Dunis (ed.), Forecasting Financial Markets. Wiley
[not found]