nonparametric neural network estimation of lyapunov exponents and a direct test for chaos this paper derives the asymptotic distribution of the nonparametric neural network estimator of the lyapunov exponent in a noisy system positivity of the lyapunov exponent is an operational definition of chaos we introduce a statistical framework for testing the chaotic hypothesis based on the estimated lyapunov exponents and a consistent variance estimator a simulation study to evaluate small sample performance is reported we also apply our procedures to daily stock return data in most cases the hypothesis of chaos in the stock return series is rejected at the 1 level with an exception in some higher power transformed absolute returns

a single blind controlled competition among tests for nonlinearity and chaos interest has been growing in testing for nonlinearity or chaos in economic data but much controversy has arisen about the available results this paper explores the reasons for these empirical difficulties we designed and ran a single blind controlled competition among five highly regarded tests for nonlinearity or chaos with ten simulated data series the data generating mechanisms include linear processes chaotic recursions and nonchaotic stochastic processes and both large and small samples were included in the experiment the data series were produced in a single blind manner by the competition manager and sent by e mail without identifying information to the experiment participants each such participant is an acknowledged expert in one of the tests and has a possible vested interest in producing the best possible results with that one test the results of this competition provide much surprising information about the power functions of some of the best regarded tests for nonlinearity or noisy chaos

the stability of electricity prices estimation and inference of the lyapunov exponents abstract the aim of this paper is to illustrate how the stability of a stochastic dynamic system is measured using the lyapunov exponents specifically we use a feedforward neural network to estimate these exponents as well as asymptotic results for this estimator to test for unstable chaotic dynamics the data set used is spot electricity prices from the nordic power exchange market nord pool and the dynamic system that generates these prices appears to be chaotic in one case since the null hypothesis of a non positive largest lyapunov exponent is rejected at the 1 per cent level

likelihood based estimation of latent generalized arch structures garch models are commonly used as latent processes in econometrics financial economics and macroeconomics yet no exact likelihood analysis of these models has been provided so far in this paper we outline the issues and suggest a markov chain monte carlo algorithm which allows the calculation of a classical estimator via the simulated em algorithm or a bayesian solution in o t computational operations where t denotes the sample size we assess the performance of our proposed algorithm in the context of both artificial examples and an empirical application to 26 uk sectorial stock returns and compare it to existing approximate solutions

fundamental limitations for estimating dimensions and lyapunov exponents in dynamical systems we show that values of the correlation dimension estimated over a decade from the grassberger procaccia algorithm cannot exceed the value 2 log10n if n is the number of points in the time series when this bound is saturated it is thus not legitimate to conclude that low dimensional dynamics is present the estimation of lyapunov exponents is also discussed

marginalization and contemporaneous aggregation in multivariate garch processes abstract we first of all show that contemporaneous aggregation of independent univariate garch processes yields a weak garch process as introduced by drost and nijman 1993 subsequently we analyze the dependence of the parameters in the aggregate on the parameters in the underlying models and show that the variance parameters after aggregation depend on the underlying variance and kurtosis parameters then we generalize the results by showing that a linear combination of variables generated by a multivariate garch process will also be weak garch finally we derive the marginal weak garch processes implied by several multivariate garch processes

the topological invariance of lyapunov exponents in embedded dynamics abstract there are two contributions of this paper one is to provide a theoretical ground for the jacobian methods by showing that n of the lyapunov exponents of the estimated jacobian are the lyapunov exponents of an n dimensional unknown dynamical system the second is to show that it is necessary that the lyapunov exponents be calculated in the tangent space of the attractor in the embedding the method of reconstruction is to form vectors of m consecutive observations which for m 2 n is generically an embedding this is takens result a function of m variables is then fit to the data and the jacobian matrix is constructed at each point in the orbit of the data when embedding occurs at dimension m n then the lyapunov exponents of the reconstructed dynamics are the lyapunov exponents of the original dynamics this is the case for the henon the lorenz and the mackey glass systems with a first component observer however if embedding only occurs for an m n then the jacobian method yields m lyapunov exponents only n of which are the lyapunov exponents of the original system the problem is that as currently used the jacobian method is applied to the full m dimensional space of the reconstruction and not just to the n dimensional manifold that is the image of the embedding map our examples show that it is possible to get spurious lyapunov exponents that are even larger than the largest lyapunov exponent of the original system

asymptotic filtering theory for univariate arch models this paper builds on this earlier work by deriving the asymptotic distribution of the measurement error this allows us to approximate the measurement accuracy of arch conditional variance estimates and compare the efficiency achieved by different arch models we are also able to characterize the relative importance of different kinds of misspecification for example we show that misspecifying conditional means adds only trivially at least asymptotically to measurement error while other factors for example capturing the leverage effect accommodating thick tailed residuals and correctly modelling the variability of the conditional variance process are potentially much more important third we are able to characterize a class of asymptotically optimal arch conditional variance estimates

a statistical framework for testing chaotic dynamics via lyapunov exponents abstract one common criticism of the existing lyapunov exponents algorithms is the absence of a distributional theory which provides a framework for the statistical hypothesis testing for the calculated lyapunov exponents this paper presents a methodology to calculate the empirical distributions of lyapunov exponents by using a bootstrapping technique the methodology of the paper provides a formal test of chaos under the null hypothesis the numerical examples show that the method works well with small data sets

minimal polynomials of integral symmetric matrices abstract we investigate what the possible minimal polynomials are for integral symmetric matrices we show that the obvious necessary conditions are sufficient for polynomials of degree at most 4 we show that necessary conditions are sufficient for minimal polynomials of self adjoint operators on positive definite unimodular lattices we also give a relatively elementary proof of the result of estes that any total real algebraic integer is the eigenvalue of an integral symmetric matrix this question was asked by alan hoffman who also showed that this result implies that any totally real algebraic integer is the eigenvalue of the adjacency matrix of some graph

the predictive content of cboe crude oil volatility index abstract volatility forecasting is an important issue in the area of econophysics the information content of implied volatility for financial return volatility has been well documented in the literature but very few studies focus on oil volatility in this paper we show that the cboe crude oil volatility index ovx has predictive ability for spot volatility of wti and brent oil returns from both in sample and out of sample perspectives including ovx based implied volatility in garch type volatility models can improve forecasting accuracy most of time the predictability from ovx to spot volatility is also found for longer forecasting horizons of 5 days and 20 days the simple garch 1 1 and fractionally integrated garch with ovx performs significantly better than the other ovx models and all 6 univariate garch type models without ovx robustness test results suggest that ovx provides different information from as short term interest rate

forecasting volatility in shanghai and shenzhen markets based on multifractal analysis abstract this paper analyzes the multifractality in shanghai and shenzhen stock markets using multifractal spectrum analysis and multifractal detrended fluctuation analysis we find that the main source of multifractality is long range correlations of large and small fluctuations then we introduce a multifractal volatility measure mv and find that by taking mv as daily conditional volatility the simulated series displayed similar stylized facts to the original daily return series by capturing the dynamics of mv using the arfima model we find that the out of sample forecasting performance of the arfima mv model is better than some garch class models and the arfima rv model under some criteria of loss function

an attractor in a solar time series a time series analysis of observed solar radio pulsations suggests that there must be a low dimensional attractor the power spectrum cannot be interpreted as a superposition of periodic components estimates of the maximum lyapunov exponent and of the kolmogorov entropy give some indications for a deterministic chaos in order to study the limitations inherent in small data samples we include data from the lorenz model and an artificial noise record consequences for the physical modelling of the pulsation event are discussed

trace of totally positive algebraic integers and integer transfinite diameter the diameter of a totally real algebraic integer of degree d with conjugates 1 2 d is diam d 1 for all positive integers k and n diam 2 cos 2k n is less than 4 r m robinson has computed modulo integer translations all the other totally real algebraic integers with diam 4 for d 8 we have done the computations for all d 15 we use a large family of explicit auxiliary functions related to generalized integer transfinite diameter of real intervals they give good bounds for the coefficients of the minimal polynomial of for d 15 we prove a recent conjecture of capparelli et al mathematics subject classification 2000 primary 11c08 11y40

nonlinearity in high frequency stock returns evidence from the athens stock exchange abstract this study investigates empirically the presence of nonlinearities in the athens composite share price index high frequency returns a preliminary analysis indicates that volatility exhibits a periodic intraday inverse j shaped pattern associated with the opening and closing of the market periodicity is then removed employing a flexible fourier form subsequently an arma figarch model over several frequencies yields that return volatility is long memory and self similar nonlinear analysis with the use of the embedding dimension suggests that the filtered return process does not exhibit deterministic or higher order stochastic nonlinearity rather it is reminiscent of a random process we conclude that the acspi data are nonlinear however nonlinearity is attributed to persistent arch effects

noisy chaos in intraday financial data evidence from the american index the presence of chaos in financial markets was inconclusive due mainly to test misspecification and data type although noisy chaos was investigated in recent studies it was only explored in daily returns which does not necessary mean that continuous intra daily data will exhibit the same dynamics high level noisy chaos is tested in the standard poor s 500 index returns over 4 different frequencies weekly daily 30 min and 5 min basis the dynamics in all frequencies are non chaotic

filtering and forecasting with misspecified arch models i getting the right variance with the wrong model this paper investigates the properties of the conditional covariance estimates generated by a misspecified arch model for example suppose that we observe a diffusion process at discrete time intervals of length h for each h we use a garch 1 1 model to estimate the instantaneous conditional covariance matrix of the diffusion under mild regularity conditions the difference between these conditional covariance estimates and the true conditional covariance converges to zero in probability as h 0 many other arch models for example exponential arch have similar consistency properties this may well account for the success of arch models in short term forecasting using high frequency data since even misspecified arch models can produce good estimates of volatility

arch models as diffusion approximations abstract this paper investigates the convergence of stochastic difference equations e g arch to stochastic differential equations as the length of the discrete time intervals between observations goes to zero these results are applied to the garch 1 1 model of bollerslev 1986 and to the ar 1 exponential arch model of nelson 1989 in their continuous time limits the conditional variance processes in these models have stationary distributions that are inverted gamma and lognormal respectively in addition a class of diffusion approximations based on the exponential arch model is developed

a multiple regime smooth transition heterogeneous autoregressive model for long memory and asymmetries in this paper we propose a flexible model to describe nonlinearities and long range dependence in time series dynamics the new model is a multiple regime smooth transition extension of the heterogeneous autoregressive har model which is specifically designed to model the behavior of the volatility inherent in financial time series the model is able to simultaneously approximate long memory behavior as well as describe sign and size asymmetries a sequence of tests is developed to determine the number of regimes and an estimation and testing procedure is presented monte carlo simulations evaluate the finite sample properties of the proposed tests and estimation procedures we apply the model to several dow jones industrial average index stocks using transaction level data from the trades and quotes database that covers ten years of data we find strong support for long memory and both sign and size asymmetries furthermore the new model when combined with the linear har model is viable and flexible for purposes of forecasting volatility

relationship between efficiency and predictability in stock price change abstract in this study we evaluate the relationship between efficiency and predictability in the stock market the efficiency which is the issue addressed by the weak form efficient market hypothesis is calculated using the hurst exponent and the approximate entropy apen the predictability corresponds to the hit rate this is the rate of consistency between the direction of the actual price change and that of the predicted price change as calculated via the nearest neighbor prediction method we determine that the hurst exponent and the apen value are negatively correlated however predictability is positively correlated with the hurst exponent

statistical tests for deterministic effects in broad band time series we derive a normalized version of the indicators of savit and green and prove that these normalized statistics have asymptotically a normal distribution with a mean of zero and standard deviation of one if the time series is random in the sense of being lid independent and identically distributed we verify this result numerically and study the magnitude of the finite size effects we also show that these statistics are very sensitive to the existence of deterministic effects in the series even if the underlying deterministic structure is complex such as those generated by a chaotic system we show that with moderate amounts of data the statistics can easily indicate the presence of an underlying attractor even in the presence of iid noise which is as large as or greater than the signal finally we discuss the generalization of our approach to include 1 other null hypotheses besides lid which express assumptions of specific dependencies and 2 the study of deterministic effects between more than one time series

long memory continuous time models this paper presents a new family of long memory models the continuous time moving average fractional process the continuous time framework allows to reconcile two competitive types of modelling fractional integration of arma processes and fractional brownian motion a comparison with usual discrete time arfima models is lead some well known empirical evidence on macroeconomic and financial time series such as variability of forward rates aggregation of responses across heterogeneous agents are well captured by this continuous time modelling moreover the usual statistical tools for long memory series and for stochastic differential equations can be jointly applied in this setting

on estimation of a regression model with long memory stationary errors we consider estimation of a regression model with long memory stationary errors first we estimate the regression parameters by the least squares estimator lse and next those describing the correlation structure of the error terms by using the residuals obtained from the lse certain regularity conditions introduced to develop the asymptotic theory no longer hold in this model in this situation we derive asymptotic properties of the preceding estimation procedure

generalized synchronization in fractional order systems the generalized synchronization of fractional order systems is investigated including the synchronization between two different fractional order systems with the same order the synchronization between two different fractional order systems with no orders the same and the synchronization between a classical system and its corresponding fractional order system with mismatched parameters the mechanism for the occurrence of generalized synchronization of fractional order systems is clarified the necessary and sufficient conditions are given and several methods to detect generalized synchronization are discussed the relationship between generalized synchronization and the equivalence of fractional order systems is also considered

the asymptotic distribution of nonparametric estimates of the lyapunov exponent for stochastic time series this paper derives the asymptotic distribution of a smoothing based estimator of the lyapunov exponent for a stochastic time series under two general scenarios in the first case we are able to establish root t consistency and asymptotic normality while in the second case which is more relevant for chaotic processes we are only able to establish asymptotic normality at a slower rate of convergence we provide consistent confidence intervals for both cases we apply our procedures to simulated data

a comment on measuring the hurst exponent of financial time series abstract a fundamental hypothesis of quantitative finance is that stock price variations are independent and can be modeled using brownian motion in recent years it was proposed to use rescaled range analysis and its characteristic value the hurst exponent to test for independence in financial time series theoretically independent time series should be characterized by a hurst exponent of 1 2 however finite brownian motion data sets will always give a value of the hurst exponent larger than 1 2 and without an appropriate statistical test such a value can mistakenly be interpreted as evidence of long term memory we obtain a more precise statistical significance test for the hurst exponent and apply it to real financial data sets our empirical analysis shows no long term memory in some financial returns suggesting that brownian motion cannot be rejected as a model for price dynamics

the method of simulated quantiles we introduce an inference method based on quantiles matching which is useful for situations where the density function does not have a closed form but it is simple to simulate and or moments do not exist functions of theoretical quantiles which depend on the parameters of the assumed probability law are matched with sample quantiles which depend on observations since the theoretical quantiles may not be available analytically the optimization is based on simulations we illustrate the method with the estimation of stable distributions a thorough monte carlo study and an illustration to 22 nancial indexes show

long term memory in stock market prices a test for long term memory that is robust to short range dependence is developed it is a modification of the r s statistic and the relevant asymptotic sampling theory is derived via functional central limit theory contrary to previous findings when applied to daily and monthly stock returns indexes over several time periods this test yields no evidence of long range dependence once short range dependence is accounted for monte carlo experiments show that the modified r s test has power against at least two specific models of long term memory suggesting that models with short range dependence may adequately capture the behavior of historical stock returns copyright 1991 by the econometric society

fractionally integrated process for transition economics we analyze the european transition economics and show that many time series of major indices exhibit i power law correlations in their values ii power law correlations in their magnitudes and iii an asymmetric probability distribution applying the phase randomization procedure to these time series we show that magnitude correlations completely vanish we propose a stochastic model that can generate time series with features i ii and iii and we show by means of numerical simulations that this model is capable of reproducing these three features found in the empirical data

indirect estimation of arfima and varfima models indirect estimation methods are proposed for estimating arfima as well as more complex varfima models a general framework for conducting indirect estimation of fractional models is developed that covers simulation methods choice of auxiliary model and estimation algorithm special attention is given to comparing the finite sampling properties of the indirect estimator with sowell s 1992a exact time domain maximum likelihood estimator the spectral maximum likelihood estimator of fox and taqqu 1986 and the geweke and porter hudak 1983 spectral regression estimator the indirect estimator can be computationally faster than the exact time domain maximum likelihood estimator while generating similar small sample properties the computational gains of the indirect estimator over maximum likelihood increase as the complexity of the data generating process increases

probability of local bifurcation type from a fixed point a random matrix perspective results regarding probable bifurcations from fixed points are presented in the context of general dynamical systems real random matrices time delay dynamical systems companion matrices and a set of mappings known for their properties as universal approximators neural networks the eigenvalue spectrum is considered both numerically and analytically using previous work of edelman et al based upon the numerical evidence various conjectures are presented the conclusion is that in many circumstances most bifurcations from fixed points of large dynamical systems will be due to complex eigenvalues nevertheless surprising situations are presented for which the aforementioned conclusion does not hold e g real random matrices with gaussian elements with a large positive mean and finite variance

a trace bound for positive definite connected integer symmetric matrices abstract let a be a connected integer symmetric matrix i e a a i j m n z for some n a a t and the underlying graph vertices corresponding to rows with vertex i joined to vertex j if a i j 0 is connected we show that if all the eigenvalues of a are strictly positive then tr a 2 n 1 there are two striking corollaries first the analogue of the schur siegel smyth trace problem is solved for characteristic polynomials of connected integer symmetric matrices second we find new examples of totally real separable irreducible monic integer polynomials that are not minimal polynomials of integer symmetric matrices

determining lyapunov exponents from a time series we present the first algorithms that allow the estimation of non negative lyapunov exponents from an experimental time series lyapunov exponents which provide a qualitative and quantitative characterization of dynamical behavior are related to the exponentially fast divergence or convergence of nearby orbits in phase space a system with one or more positive lyapunov exponents is defined to be chaotic our method is rooted conceptually in a previously developed technique that could only be applied to analytically defined model systems we monitor the long term growth rate of small volume elements in an attractor the method is tested on model systems with known lyapunov spectra and applied to data for the belousov zhabotinskii reaction and couette taylor flow

scaling symmetry renormalization and time series modeling the case of financial assets dynamics we present and discuss a stochastic model of financial assets dynamics based on the idea of an inverse renormalization group strategy with this strategy we construct the multivariate distributions of elementary returns based on the scaling with time of the probability density of their aggregates in its simplest version the model is the product of an endogenous autoregressive component and a random rescaling factor designed to embody also exogenous influences mathematical properties like increments stationarity and ergodicity can be proven thanks to the relatively low number of parameters model calibration can be conveniently based on a method of moments as exemplified in the case of historical data of the s p500 index the calibrated model accounts very well for many stylized facts like volatility clustering power law decay of the volatility autocorrelation function and multiscaling with time of the aggregated return distribution in agreement with empirical evidence in finance the dynamics is not invariant under time reversal and with suitable generalizations skewness of the return distribution and leverage effects can be included the analytical tractability of the model opens interesting perspectives for applications for instance in terms of obtaining closed formulas for derivative pricing further important features are the possibility of making contact in certain limits with autoregressive models widely used in finance and the possibility of partially resolving the long and short memory components of the volatility with consistent results when applied to historical series

estimating the intrinsic dimension of data with a fractal based method in this paper the problem of estimating the intrinsic dimension of a data set is investigated a fractal based approach using the grassberger procaccia algorithm is proposed since the grassberger procaccia algorithm 1983 performs badly on sets of high dimensionality an empirical procedure that improves the original algorithm has been developed the procedure has been tested on data sets of known dimensionality and on time series of santa fe competition

a goodness of fit test for arch models a goodness of fit test in the class of conditional heteroscedastic time series models is examined due to the nonstandard limiting distribution of the test we propose to bootstrap the test showing its asymptotic validity moreover we illustrate the finite sample performance of the test by a small monte carlo study

testing for nonlinearity in time series the method of surrogate data we describe a statistical approach for identifying nonlinearity in time series the method first specifies some linear process as a null hypothesis then generates surrogate data sets which are consistent with this null hypothesis and finally computes a discriminating statistic for the original and for each of the surrogate data sets if the value computed for the original data is significantly different than the ensemble of values computed for the surrogate data then the null hypothesis is rejected and nonlinearity is detected we discuss various null hypotheses and discriminating statistics the method is demonstrated for numerical data generated by known chaotic systems and applied to a number of experimental time series which arise in the measurement of superfluids brain waves and sunspots we evaluate the statistical significance of the evidence for nonlinear structure in each case and illustrate aspects of the data which this approach identifies

sup norm approximation bounds for networks through probabilistic methods we consider the problem of approximating a smooth target function and its derivatives by networks involving superpositions and translations of a fixed activation function the approximation is with respect to the sup norm and the rate is shown to be of order o n sup 1 2 that is the rate is independent of the dimension d the results apply to neural and wavelet networks and extend the work of barren see proc 7th yale workshop on adaptive and learning systems may 1992 and ibid vol 39 p 930 1993 the approach involves probabilistic methods based on central limit theorems for empirical processes indexed by classes of functions

a tight upper bound on the gain of linear and nonlinear predictors for stationary stochastic processes one of the striking questions in prediction theory is this is there a chance to predict future values of a given signal usually we design a predictor for a special signal or problem and then measure the resulting prediction quality if there is no a priori knowledge on the optimal predictor the achieved prediction gain will depend strongly of the prediction model used to cope with this lack of knowledge a theorem on the maximum achievable prediction gain of stationary signals is presented this theorem provides the foundation for estimating a quality goal for the predictor design independent of a special predictor implementation linear or nonlinear as usual the prediction gain is based on the mean square error mse of the predicted signal the achievable maximum of the prediction gain is calculated using an information theoretic quantity known as the mutual information in order to obtain the gain we use a nonparametric approach to estimate the maximum prediction gain based on the observation of one specific signal we illustrate this by means of well known example signals and show an application to load forecasting an estimation algorithm for the prediction gain has been implemented and used in the experimental part of the paper

characterization of strange attractors a new measure of strange attractors is introduced which offers a practical algorithm to determine their character from the time series of a single observable the relation of this new measure to fractal dimension and information theoretic entropy is discussed

time series analysis of s p 500 index a horizontal visibility graph approach the behavior of stock prices has been thoroughly studied throughout the last century and contradictory results have been reported in the corresponding literature in this paper a network theoretical approach is provided to investigate how crises affected the behavior of us stock prices we analyze high frequency data from s p500 via the horizontal visibility graph method and find that all major crises that took place worldwide in the last twenty years affected significantly the behavior of the price index nevertheless we observe that each of those crises impacted the index in a different way and magnitude interestingly our results suggest that the predictability of the price index series increases during the periods of crises

highly flexible distributions to fit multiple frequency financial returns abstract financial data are usually studied via low flexible distributions independently of the frequency of the data due to their simplicity and analytical tractability in this paper we analyze two highly flexible five parameter distributions into fitting financial returns these are the skewed generalized t sgt and the generalized hyperbolic gh applications carried on two exchange rates euro dollar and dollar yen and two indexes s p 500 and nikkei 225 over four frequencies weekly daily 30 min and 5 min confirm the superiority of the sgt and gh in approximating the distribution of a given data at a remarkable precision moreover as we move from higher to lower frequency the distribution s overall shape does indeed change radically and the estimated parameters refute the tendency to normality which calls into question the aggregational gaussianity s stylized fact

the fractional unit root distribution asymptotic distributions are derived for the ordinary least squares estimate of a first order autoregression model when the series is fractionally integrated the fractional unit root distribution is introduced to describe the limiting distribution the unit root distribution is shown to be an atypical member of this family because its density is nonzero over the entire real line for 1 2 copyright 1990 by the econometric society

density estimation under long range dependence dehling and taqqu established the weak convergence of the empirical process for a long range dependent stationary sequence under gaussian subordination we show that the corresponding density process based on kernel estimators of the marginal density converges weakly with the same normalization to the derivative of their limiting process the phenomenon which carries on for higher derivatives and for functional laws of the iterated logarithm is in contrast with independent or weakly dependent situations where the density process cannot be tight in the usual function spaces with supremum distances

modeling long memory in stock market volatility inspired by the idea that regime switching may give rise to persistence that is observationally equivalent to a unit root we derive a regime switching process that exhibits long memory the feature of the process that generates long memory is a heavy tailed duration distribution using this process for volatility we obtain a regime switching stochastic volatility rssv model that we fit to daily s p returns from 1928 through 1995 by means of the efficient method of moments estimation emm method forecasts of rssv volatility given past returns can be generated by reprojection as we illustrate the rssv model is accepted according to the emm chi squared statistic using this statistic we also evaluate several other models that have been proposed in the literature and some modifications to them we find that models that exhibit long memory in volatility and heavy tails conditionally as does the rssv model fit the data whereas models without these characteristics do not we also find weak evidence that suggests the presence of an additional short memory component of volatility over and above the long memory component

analysis of shares frequency components on daily value at risk in emerging and developed markets abstract value at risk was estimated using the technique of wavelet decomposition with the goal to analyze the frequency components impacts on variances of daily stock returns and var forecasts daily returns of twenty one shares of the ibovespa and daily returns of twenty two shares of the djia were used the figarch 1 d 1 model was applied to the reconstructed returns to model and establish the prediction of conditional variance applying the rolling window technique the value at risk was then estimated and the results showed that the djia shares showed more efficient market behavior than those of ibovespa the differences in behavior induce to affirm that vars used in the analysis of financial assets from different markets with different governance premises should be estimated by series of returns reconstructed by aggregations of components of different frequencies a set of back testing was applied to confront the estimated vars which demonstrated that the estimations of vars models are consistent

assessing nonlinear structures in real exchange rates using recurrence plot strategies purchasing power parity ppp is an important theory at the basis of a large number of economic models however the implication derived from the theory that real exchange rates must follow stationary processes is not conclusively supported by empirical studies in a recent paper serletis and gogas appl finance econ 10 2000 615 show evidence of deterministic chaos in several oecd exchange rates as a consequence ppp rejections could be spurious in this work we follow a two stage testing procedure to test for nonlinearities and chaos in real exchange rates using a new set of techniques designed by webber and zbilut j appl physiol 76 1994 965 called recurrence quantification analysis rqa our conclusions differ slightly from serletis and gogas appl finance econ 10 2000 615 but they are also supportive of chaos for some exchange rates

the trace of totally positive algebraic integers for all totally positive algebraic numbers except a finite number of explicit exceptions the following inequality holds 1 d 1 d max 1 780022 1 66 1 where d is the degree of a and 0 1 d its conjugates this improves previous results of smyth flammang and rhin

properties of a simple bilinear stochastic model estimation and predictability we analyze the properties of arguably the simplest bilinear stochastic multiplicative process proposed as a model of financial returns and of other complex systems combining both nonlinearity and multiplicative noise by construction it has no linear predictability zero two point correlation but a certain nonlinear predictability non zero three point correlation it can thus be considered as a paradigm for testing the existence of a possible nonlinear predictability in a given time series we present a rather exhaustive study of the process including its ability to produce fat tailed distributions from gaussian innovations the unstable characteristics of the inversion of the key nonlinear parameters and of the two initial conditions necessary for the implementation of a prediction scheme and an analysis of the associated super exponential sensitivity of the inversion of the innovations in the presence of a large impulse our study emphasizes the conditions under which a degree of predictability can be achieved and describe a number of different attempts which overall illuminates the properties of the process in conclusion notwithstanding its remarkable simplicity the bilinear stochastic process exhibits remarkably rich and complex behavior which makes it a serious candidate for the modeling of financial time series among others

fractionally integrated generalized autoregressive conditional heteroskedasticity abstract the new class of fractionally integrated generalized autoregressive conditionally heteroskedastic figarch processes is introduced the conditional variance of the process implies a slow hyperbolic rate of decay for the influence of lagged squared innovations unlike i d processes for the mean maximum likelihood estimates mle of the figarch parameters are argued to be t 1 2 consistent the small sample behavior of an approximate mle procedure is assessed through a simulation study which also documents how the estimation of a standard garch model tends to produce integrated or igarch like estimates an empirical example with daily deutschmark u s dollar exchange rates illustrates the practical relevance of the new figarch specification

unbiased determination of production technologies abstract to determine whether an industry exhibits constant returns to scale whether the production function is homothetic or whether inputs are separable a common approach is to specify a cost function estimate its parameters using data such as prices and quantities of inputs and then test the parametric restrictions corresponding to constant returns a homothetic technology or separability statistically such inferences are valid if the true cost function is a member of the parametric class considered otherwise the inference is biased that is the true rejection probability is not necessarily adequately approximated by the nominal size of the statistical test the use of fixed parameter flexible functional forms such as the translog the generalized leontief or the box cox will not alleviate this problem the fourier flexible form differs fundamentally from other flexible forms in that it has a variable number of parameters and a known bound depending on the number of parameters on the error as measured by the sobolev norm of approximation to an arbitrary cost function thus it is possible to construct statistical tests for constant returns a homothetic technology or separability which are asymptotically size by letting the number of parameters of the fourier flexible form depend on sample size that is the true rejection probability converges to the nominal size of the test as sample size tends to infinity the rate of convergence depends on the smoothness of the true cost function the more times is differentiable the true cost function the faster the convergence the method is illustrated using the data on aggregate u s manufacturing of berndt and wood 1975 1979 and berndt and khaled 1979

the law of the iterated logarithm for self similar processes represented by multiple wiener integrals on etablit une loi du logarithme itere fonctionnelle pour certains processus autosimilaires qui sont des images applications continues du mouvement brownien et on obtient le resultat desire en utilisant l approximation par de tels processus

testing for strong serial correlation and dynamic conditional heteroskedasticity in multiple regression abstract broad classes of diagnostics for serial correlation and or dynamic conditional heteroskedasticity of regression disturbances are considered the classes include statistics with good power against strongly dependent alternatives along with the usual ones that test against weak dependence and many others limiting null distributions are obtained under mild conditions on the dependence structure of the alternative against which the test is derived on moments of the disturbances and on the regressors the various test statistics have a similar overall structure and while tests against strongly dependent alternatives entail more computation than ones against weakly dependent alternatives the difference can be slight if the fast fourier transform is used

notes on financial econometrics the first part of the discussion reviews recent successes in modeling of discrete time financial data and argues that a direct approach is better suited than stochastic volatility the second part reviews recent work on estimating continuous time models with emphasis on simulation based techniques and joint estimation of the risk neutral and objective probability distributions

stochastic garch dynamics describing correlations between stocks the arch and garch processes have been successfully used for modelling price dynamics such as stock returns or foreign exchange rates analysing the long range correlations between stocks we propose a model based on the garch process which is able to describe the main characteristics of the stock price correlations including the mean variance probability density distribution and the noise spectrum

generalized autoregressive conditional heteroskedasticity a natural generalization of the arch autoregressive conditional heteroskedastic process introduced in engle 1982 to allow for past conditional variances in the current conditional variance equation is proposed stationarity conditions and autocorrelation structure for this new class of parametric models are derived maximum likelihood estimation and testing are also considered finally an empirical example relating to the uncertainty of the inflation rate is presented

a practical method for calculating largest lyapunov exponents from small data sets detecting the presence of chaos in a dynamical system is an important problem that is solved by measuring the largest lyapunov exponent lyapunov exponents quantify the exponential divergence of initially close state space trajectories and estimate the amount of chaos in a system we present a new method for calculating the largest lyapunov exponent from an experimental time series the method follows directly from the definition of the largest lyapunov exponent and is accurate because it takes advantage of all the available data we show that the algorithm is fast easy to implement and robust to changes in the following quantities embedding dimension size of data set reconstruction delay and noise level furthermore one may use the algorithm to calculate simultaneously the correlation dimension thus one sequence of computations will yield an estimate of both the level of chaos and the system complexity

the functional iterated logarithm law for stochastic processes represented by multiple wiener integrals in a previous paper the authors obtained a functional law of the iterated logarithm for a class of self similar processes bar x with stationary increments which are represented by multiple wiener integrals this result is extended to a certain class of processes represented by multiple wiener integrals which converge to bar x with an appropriate normalization as an application a functional log log law for nonlinear functionals of some stationary gaussian processes is given

computing totally positive algebraic integers of small trace we construct minimal polynomials of totally positive algebraic integers of small absolute trace by consideration of their reductions modulo auxiliary polynomials many new examples of such polynomials of minimal absolute trace for given degree are found the computations are pushed to degrees that previously were unattainable and one consequence is that the new examples form the majority of all those known as an application we produce a new bound for the schur siegel smyth trace problem

investigating detrended fluctuation analysis with structural breaks abstract detrended fluctuation analysis has been used in several fields of science to study the statistical properties of trend stationary and nonstationary time series its application to financial data has produced important results concerning long range correlations and long memory however these results may be contaminated if the researcher attributes to nonstationary trends the effect of stationary trends with endogenous structural breaks our paper proposes a modified dfa model where boxes to determine local trends are replaced by endogenous structural break windows we also allow local trends fitted by quadratic functions and use squared residuals in place of patchy standard deviations to study the magnitude of the power law exponent the results show that our modified dfa model performs better than the fixed length alternatives originally proposed and is therefore a suitable model to fit with financial data consistently with previous findings our results show positive long range correlation in all indices with the higher value for emerging markets

on maximum likelihood estimation of the differencing parameter of fractionally integrated noise with unknown mean abstract there are two approaches to maximum likelihood ml estimation of the parameter of fractionally integrated noise approximate frequency domain ml fox and taqqu 1986 and exact time domain ml sowell 1992b if the mean of the process is known then a clear finite sample mean squared error ranking of the estimators emerges the exact time domain estimator is superior we show in this paper however that the finite sample efficiency of approximate frequency domain ml relative to exact time domain ml rises dramatically when the mean is unknown and so must be estimated the intuition for our result is straightforward the frequency domain ml estimator is invariant to the true but unknown mean of the process while the time domain ml estimator is not feasible time domain estimation must therefore be based upon de meaned data but the long memory associated with fractional integration makes precise estimation of the mean difficult we conclude that the frequency domain estimator is an attractive and efficient alternative for situations in which large sample sizes render time domain estimation impractical

an algorithm for the n lyapunov exponents of an n dimensional unknown dynamical system abstract an algorithm for estimating lyapunov exponents of an unknown dynamical system is designed the algorithm estimates not only the largest but all lyapunov exponents of the unknown system the estimation is carried out by a multivariate feedforward network estimation technique we focus our attention on deterministic as well as noisy system estimation the performance of the algorithm is very satisfactory in the presence of noise as well as with limited number of observations

temporal aggregation of garch processes the authors derive low frequency say weekly models implied by high frequency say daily arma models with symmetric garch errors they show that low frequency models exhibit conditional heteroskedasticity of the garch form as well the parameters in the conditional variance equation of the low frequency model depend upon mean variance and kurtosis parameters of the corresponding high frequency model moreover strongly consistent estimators of the parameters in the high frequency model can be derived from low frequency data the common assumption in applications that rescaled innovations are independent is disputable since it depends upon the available data frequency copyright 1993 by the econometric society

consistency and asymptotic normality of the quasi maximum likelihood estimator in igarch 1 1 and covariance stationary garch 1 1 models this paper provides a proof of the consistency and asymptotic normality of the quasi maximum likelihood estimator in garch 1 1 and igarch 1 1 models in contrast to the case of a unit root in the conditional mean the presence of a unit root in the conditional variance does not affect the limiting distribution of the estimators in both models estimators are normally distributed in addition a consistent estimator of the covariance matrix is available enabling the use of standard test statistics for inference

time dependent hurst exponent in financial time series abstract we calculate the hurst exponent h t of several time series by dynamical implementation of a recently proposed scaling technique the detrending moving average dma in order to assess the accuracy of the technique we calculate the exponent h t for artificial series simulating monofractal brownian paths with assigned hurst exponents h we next calculate the exponent h t for the return of high frequency tick by tick sampled every minute series of the german market we find a much more pronounced time variability in the local scaling exponent of financial series compared to the artificial ones the dma algorithm allows the calculation of the exponent h t without any a priori assumption on the stochastic process and on the probability distribution function of the random variables as happens for example in the case of the kitagawa grid and the extended kalmann filtering methods the present technique examines the local scaling exponent h t around a given instant of time this is a significant advance with respect to the standard wavelet transform or to the higher order power spectrum technique which instead operate on the global properties of the series by legendre or fourier transform of qth order moments

multiscale local change point detection with applications to value at risk this paper offers a new approach to modeling and forecasting of non stationary time series with applications to volatility modeling for financial data the approach is based on the assumption of local homogeneity for every time point there exists a historical interval of homogerzeity in which the volatility parameter can be well approximated by a constant the proposed procedure recovers this interval from the data using the local change point lcp analysis afterward the estimate of the volatility can be simply obtained by local averaging the approach carefully addresses the question of choosing the tuning parameters of the procedure using the so called propagation condition the main result claims a new oracle inequality in terms of the modeling bias which measures the quality of the local constant approximation this result yields the optimal rate of estimation for smooth and piecewise constant volatility functions then the new procedure is applied to some data sets and a comparison with a standard garch model is also provided finally we discuss applications of the new method to the value at risk problem the numerical results demonstrate a very reasonable performance of the new method

consistent ranking of volatility models abstract we show that the empirical ranking of volatility models can be inconsistent for the true ranking if the evaluation is based on a proxy for the population measure of volatility for example the substitution of a squared return for the conditional variance in the evaluation of arch type models can result in an inferior model being chosen as best with a probability that converges to one as the sample size increases we document the practical relevance of this problem in an empirical application and by simulation experiments our results provide an additional argument for using the realized variance in out of sample evaluations rather than the squared return we derive the theoretical results in a general framework that is not specific to the comparison of volatility models similar problems can arise in comparisons of forecasting models whenever the predicted variable is a latent variable

long term memory and volatility clustering in high frequency price changes abstract we studied the long term memory in diverse stock market indices and foreign exchange rates using detrended fluctuation analysis dfa for all high frequency market data studied no significant long term memory property was detected in the return series while a strong long term memory property was found in the volatility time series the possible causes of the long term memory property were investigated using the return data filtered by the ar 1 model reflecting the short term memory property the garch 1 1 model reflecting the volatility clustering property and the figarch model reflecting the long term memory property of the volatility time series the memory effect in the ar 1 filtered return and volatility time series remained unchanged while the long term memory property diminished significantly in the volatility series of the garch 1 1 filtered data notably there is no long term memory property when we eliminate the long term memory property of volatility by the figarch model for all data used although the hurst exponents of the volatility time series changed considerably over time those of the time series with the volatility clustering effect removed diminish significantly our results imply that the long term memory property of the volatility time series can be attributed to the volatility clustering observed in the financial time series

strong invariance principles for sequential bahadur kiefer and vervaat error processes of long range dependent sequences in this paper we study strong approximations invariance principles of the sequential uniform and general bahadur kiefer processes of long range dependent sequences we also investigate the strong and weak asymptotic behavior of the sequential vervaat process that is the integrated sequential bahadur kiefer process properly normalized as well as that of its deviation from its limiting process the so called vervaat error process it is well known that the bahadur kiefer and the vervaat error processes cannot converge weakly in the i i d case in contrast to this we conclude that the bahadur kiefer and vervaat error processes as well as their sequential versions do converge weakly to a dehling taqqu type limit process for certain long range dependent sequences

improved cost functions for modelling of noisy chaotic time series abstract standard least squares cost functions yield unbiased results only if the independent variables are noise free which is not the case in time series analysis new minimization problems for the determination of the dynamics underlying noisy chaotic data are formulated which can overcome the problem of noise in the independent variables for a given model these improved cost functions give the chance to estimate its parameters with a strongly reduced bias in the case of large amplitude measurement noise the method is illustrated by the help of numerical and experimental examples results for dynamical noise are discussed

neglecting parameter changes in garch models if a garch model is estimated on a time series that contains parameter changes in the conditional volatility process and these parameter changes are not accounted for a distinct error in the estimation occurs the sum of the estimated autoregressive parameters of the conditional variance converges to one in finite samples the sum of the estimated autoregressive parameters is heavily biased towards one this paper shows that this convergence holds for all common estimators of garch simulations of the garch model show that the effect occurs for realistic parameter changes and sample sizes for financial volatility data

out of sample forecasts of quadratic variation we compare the forecasts of quadratic variation given by the realized volatility rv and the two scales realized volatility tsrv computed from high frequency data in the presence of market microstructure noise under several different dynamics for the volatility process and assumptions on the noise we show that tsrv largely outperforms rv whether looking at bias variance rmse or out of sample forecasting ability an empirical application to all djia stocks confirms the simulation results

estimating long range dependence finite sample properties and confidence intervals abstract a major issue in financial economics is the behavior of asset returns over long horizons various estimators of long range dependence have been proposed even though some have known asymptotic properties it is important to test their accuracy by using simulated series of different lengths we test r s analysis detrended fluctuation analysis and periodogram regression methods on samples drawn from gaussian white noise the dfa statistics turns out to be the unanimous winner unfortunately no asymptotic distribution theory has been derived for this statistics so far we were able however to construct empirical i e approximate confidence intervals for all three methods the obtained values differ largely from heuristic values proposed by some authors for the r s statistics and are very close to asymptotic values for the periodogram regression method

routes to chaos in high dimensional dynamical systems a qualitative numerical study this paper examines the most probable route to chaos a high dimensional dynamical systems function space time delay neural networks endowed with a probability measure in a computational setting the most probable route to chaos relative to the measure we impose on the function space as the dimension is increased is observed to be a sequence of neimark sacker bifurcations into chaos the analysis is composed of the study of an example dynamical system followed by a probabilistic study of the ensemble of dynamical systems from which the example was drawn a scenario depicting the decoupling of the stable manifolds of the torus leading up to the onset of chaos in high dimensional dissipative dynamical systems is also presented c 2006 elsevier b v all rights reserved

maximum likelihood estimation of stationary univariate fractionally integrated time series models abstract to estimate the parameters of a stationary univariate fractionally integrated time series the unconditional exact likelihood function is derived this allows the simultaneous estimation of all the parameters of the model by exact maximum likelihood issues involved in obtaining maximum likelihood estimates are discussed particular attention is given to efficient procedures to evaluate the likelihood function obtaining starting values and the small sample properties of the estimators limitations of previous estimation procedures are also discussed

a novel approach to detect volatility clusters in financial time series abstract the self similarity index has been consolidated as a widely applied measure to quantify long memory in stock markets in this article though we shall provide a novel methodology allowing the detection of clusters of volatility in series of asset returns with this aim the concept of a volatility series is introduced we found that the existence of clusters of high low volatility in the series leads to an increasing hurst exponent of the volatility series some empirical applications were carried out in fact we artificially generated processes with clusters of volatility the mean self similarity exponents of their volatility series were compared with those from a brownian motion and the s p500 index as a result the greater the number of periods used in the calculation of the self similarity exponent the smoother the graph of the volatility series and hence the higher its exponent accordingly the more likely the clusters of volatility appear a dynamic study on the evolution of the self similarity index of the volatility series of the s p500 was also performed it was observed that the greater its self similarity exponent the more frequent the volatility changes which usually corresponds to falls in the index and hence the more likely the volatility clusters appear

binomial markov switching multifractal model with skewed t innovations and applications to chinese ssec index abstract this paper presents the binomial markov switching multifractal bmsm model of asset returns with skewed t innovations bmsm skewed t for short which considers the fat tails skewness and multifractality in asset returns simultaneously the parameters of bmsm skewed t model can be estimated by maximum likelihood ml methods and volatility forecasting can be accomplished via bayesian updating in order to evaluate the performance of bmsm skewed t model bmsm model with normal innovations bmsm n bmsm model with student t innovations bmsm t and garch 1 1 models garch n garch t and garch skewed t are chosen for comparison through empirical studies on shanghai stock exchange composite index ssec we find that for sample estimation bmsm models outperform the garch 1 1 models through bic and aic rules and bmsm skewed t performs the best among all the models due to its fat tails skewness and multifractality in addition bmsm skewed t model dominates other models at most forecasting horizons for out of sample volatility forecasts in terms of mse mae and spa test

stationarity of garch processes and of some nonnegative time series abstract two time series models are considered garch processes and generalized multivariate autoregressive equations x n 1 a n 1 x n b n 1 with nonnegative i i d coefficients in each case a necessary and sufficient condition ensuring the existence of a strictly stationary solution is given

theoretical results on fractionally integrated exponential generalized autoregressive conditional heteroskedastic processes abstract here we present a theoretical study on the main properties of fractionally integrated exponential generalized autoregressive conditional heteroskedastic fiegarch processes we analyze the conditions for the existence the invertibility the stationarity and the ergodicity of these processes we prove that if x t t z is a fiegarch p d q process then under mild conditions ln x t 2 t z is an arfima q d 0 with correlated innovations that is an autoregressive fractionally integrated moving average process the convergence order for the polynomial coefficients that describes the volatility is presented and results related to the spectral representation and to the covariance structure of both processes ln x t 2 t z and ln t 2 t z are discussed expressions for the kurtosis and the asymmetry measures for any stationary fiegarch p d q process are also derived the h step ahead forecast for the processes x t t z ln t 2 t z and ln x t 2 t z are given with their respective mean square error of forecast the work also presents a monte carlo simulation study showing how to generate estimate and forecast based on six different fiegarch models the forecasting performance of six models belonging to the class of autoregressive conditional heteroskedastic models namely arch type models and radial basis models is compared through an empirical application to brazilian stock market exchange index