Dependence structures in financial time series: a chaos-theoretic approach

Of much interest in financial econometrics is the recovery of joint distributional behaviour of collections of contemporaneous financial time series, e.g., two related commodity price series, or two asset returns series. An approach to model their joint behaviour is to use copulas. Essentially, copulas are selected on the basis of a measure of correlation between the two series and are made to match their marginal properties. Of course, generalisations exist for more than two series. A possible limitation of this approach is that only linear correlations between series might be captured. We consider incorporating more general dependence structures, through the use of the correlation integral (as in the BDS test), as a means to refine the choice of candidate copulas in an empirical situation.Archimedean copula; copula; correlation integral; dependence; Poisson convergence

Edgeworth expansion for the sample autocorrelation function

Fractionally integrated process, introduced independently by Granger and Joyeux (1980) and Hosking (1981), exhibit second-order dependence structures of rich variety, and stir much interest by way of their mathematical properties and their applications in modelling real phenomena. Their mathematical complexity oers signicant challenges in deriving estimates of parameters relating to the long memory behaviour, both in parametric and non-parametric models, with the latter having slower convergence properties. Some seminal papers include those by Yajima (1985), Fox and Taqqu (1986), and Dahlhaus (1988, 1989) on parametric estimation, and by Hurst (1951), Geweke and Porter-Hudak (1983), Robinson (1995) and Hurvich et al. (1998). We have in mind to consider the ACF bootstrap (as it is called), based on a result of Ramsey (1974), which generates a surrogate series

Binary time series generated by chaotic logistic maps

This paper examines stochastic pairwise dependence structures in binary time series obtained from discretised versions of standard chaotic logistic maps. It is motivated by applications in communications modelling which make use of so-called chaotic binary sequences. The strength of non-linear stochastic dependence of the binary sequences is explored. In contrast to the original chaotic sequence, the binary version is non-chaotic with non-Markovian non-linear dependence, except in a special case. Marginal and joint probability distributions, and autocorrelation functions are elicited. Multivariate binary and more discretised time series from a single realisation of the logistic map are developed from the binary paradigm. Proposals for extension of the methodology to other cases of the general logistic map are developed. Finally, a brief illustration of the place of chaos-based binary processes in chaos communications is given.Binary sequence; chaos; chaos communications; dependence; discretisation; invariant distribution; logistic map; randomness

Does Company Specific News Effect the US, UK, and Australian Markets within 60 minutes?

The efficient market hypothesis states that an efficient market rapidly incorporates all available information into the price of the asset. It has been well established that no market, particularly the stock market, is truly efficient as there are too many traders with differing strategies, and differing access to and interpretation of information. Despite this there is considerable evidence that the stock market does assimilate new information into prices. There has however been little research into the intraday effect of company specific news. In this paper we investigate the intraday effect of company specific news on the US, UK, and Australian markets. We use a scheme to determine whether the markets react to news by determining the likelihood of certain events occurring, and the likelihood of news occurring within 60 minutes of them, and compare the two. We find that there is strong evidence that these markets do react to news within 60 minutes, and indicate which events are most likely to correlate to news.Return; Volatility; News

Adaptive orthogonal series estimation in additive stochastic regression models

In this paper, we consider additive stochastic nonparametric regression models. By approximating the nonparametric components by a class of orthogonal series and using a generalized cross-validation criterion, an adaptive and simultaneous estimation procedure for the nonparametric components is constructed. We illustrate the adaptive and simultaneous estimation procedure by a number of simulated and real examples.Adaptive estimation; additive model; dependent process; mixing condition; nonlinear time series; nonparametric regression; orthogonal series; strict stationarity; truncation parameter

Statistical tests for Lyapunov exponents of deterministic systems.

In order to develop statistical tests for the Lyapunov exponents of deterministic dynamical systems, we develop bootstrap tests based on empirical likelihood for percentiles and expectiles of strictly stationary processes. The percentiles and expectiles are estimated in terms of asymmetric least deviations and asymmetric least squares methods. Asymptotic distributional properties of the estimators are established.

Concentration of submicrometre particles from vehicle emissions near a major road.

As part of a program of study to assess the exposure risks related to particulate matter in the outdoor environment, number concentrations of particles from vehicle emissions were measured at increasing distances from a major road. Particles in the size range from 0.015 – 0.697 µm were measured with the Scanning Mobility Particle Sizer (SMPS) and in the size range from 0.5 – 20 µm, with the Aerodynamic Particle Sizer (APS). In addition to number concentration measurements, an approximation of PM2.5 fraction was obtained using a DustTrak (simple photometer). The measurements conducted at distances from the road ranging from 15 to 375 m showed, that for conditions where the wind is blowing directly from the road, the concentration of fine and ultrafine particles decays to around half of the maximum (measured at the closest point to the road) at a distance of approximately 100 - 150 m from the road. For the wind blowing parallel to the road, the reduction to half of the concentration occurs at 50 – 100 m. There is no effect on total particle number concentration at a distance greater than 15 m from the road when the wind is blowing towards the road and away from the sampling points. Total number concentrations of larger particles measured were not significantly higher than the average values for the urban environment, and decrease with distance from the road, reaching about 60% at 150 m from the road for wind from the road. PM2.5 levels also decrease with distance to around 75% for wind from the road and to 65% for wind parallel to the road, at a distance of 375 m

Exact Maximum Likelihood estimation for the BL-GARCH model under elliptical distributed innovations

In this paper, we discuss the class of Bilinear GATRCH (BL-GARCH) models which are capable of capturing simultaneously two key properties of non-linear time series : volatility clustering and leverage effects. It has been observed often that the marginal distributions of such time series have heavy tails ; thus we examine the BL-GARCH model in a general setting under some non-Normal distributions. We investigate some probabilistic properties of this model and we propose and implement a maximum likelihood estimation (MLE) methodology. To evaluate the small-sample performance of this method for the various models, a Monte Carlo study is conducted. Finally, within-sample estimation properties are studied using S&P 500 daily returns, when the features of interest manifest as volatility clustering and leverage effects.BL-GARCH process, elliptical distribution, leverage effects, Maximum Likelihood, Monte Carlo method, volatility clustering.

Phase randomisation: a convergence diagnostic test for MCMC

Most MCMC users address the convergence problem by applying diagnostic tools to the output produced by running their samplers. Potentially useful diagnostics may be borrowed from diverse areas such as time series. One such method is phase randomisation. The aim of this paper is to describe this method in the context of MCMC, summarise its characteristics, and contrast its performance with those of the more common diagnostic tests for MCMC. It is observed that the new tool contributes information about third and higher order cumulant behaviour which is important in characterising certain forms of nonlinearity and nonstationarity.Convergence diagnostics; higher cumulants; Markov Chain Monte Carlo; non-linear time series; stationarity; surrogate series

On optimal smoothing of density estimators obtained from orthogonal polynomial expansion methods

We discuss the application of orthogonal polynomials to the estimation of probability density functions, particularly with regard to accessing features of a portfolio’s profit/loss distribution. Such expansions are given by the sum of known orthogonal polynomials multiplied by an associated weight function. However, naive applications of expansion methods are flawed. The shape of the estimator’s tail can undulate under the influence of the constituent polynomials in the expansion, and it can even exhibit regions of negative density. This paper presents techniques to remedy these flaws and improve the quality of risk estimation.We show that by targeting a smooth density that is sufficiently close to the target density, we can obtain expansion-based estimators that do not have the shortcomings of equivalent naive estimators. In particular, we apply optimization and smoothing techniques that place greater weight on the tails than on the body of the distribution. Numerical examples using both real and simulated data illustrate our approach. We further outline how our techniques can apply to a wide class of expansion methods and indicate opportunities to extend to the multivariate case, where distributions of individual component risk factors in a portfolio can be accessed for the purpose of risk management