A weak bifurcation theory for discrete time stochastic dynamical systems

This article presents a bifurcation theory of smooth stochastic dynamical systems that are governed by everywhere positive transition densities. The local dependence structure of the unique strictly stationary evolution of such a system can be expressed by the ratio of joint and marginal probability densities; this `dependence ratio' is a geometric invariant of the system. By introducing a weak equivalence notion of these dependence ratios, we arrive at a bifurcation theory for which in the compact case, the set of stable (non-bifurcating) systems is open and dense. The theory is illustrated with some simple examples.

Phenomenological and ratio bifurcations of a class of discrete time stochastic processes

Zeeman proposed a classification of stochastic dynamical systems based on the Morse classification of their invariant probability densities; the associated bifurcations are the ‘phenomenological bifurcations’ of L. Arnold. The classification is however not invariant under diffeomorphisms of the state space. In a recent paper we proposed an alternative classification, based on an invariant that is a ratio of joint and marginal probability density functions, that does not suffer from this defect. This classification entails the concept of what we call ‘ratio bifurcations’. In this note it is shown that for a large class of dynamical systems, ratio bifurcations and phenomenological bifurcations actually coincide. Moreover, we link the ratio invariant to the transformation invariant function that Wagenmakers et al. obtained for stochastic differential equations. The results are illustrated with numerical applications to stochastic dynamical systems.

The Nonlinear Dynamic Relationship of Exchange Rates: Parametric and Nonparametric Causality testing

The present study investigates the long-term linear and nonlinear causal linkages among six currencies, namely EUR/USD, GBP/USD, USD/JPY, USD/CHF, AUD/USD and USD/CAD. The prime motivation for choosing these exchange rates comes from the fact that they are the most liquid and widely traded, covering about 90% of total FX trading worldwide. The data spans two periods (PI: 3/20/1991 \u2013 3/20/1997, PII: 3/20/2003 \u2013 3/20/2007) before and after the structural break of the Asian financial crisis, which set a platform for departure for causality testing. We apply a new nonparametric test for Granger non-causality by Diks and Panchenko (2005, 2006) as well as the conventional linear Granger test on the return time series. To ensure that any causality is strictly nonlinear in nature, we also examine the nonlinear causal relationships of pairwise VAR filtered residuals as well as in a six-variate formulation. We find remaining significant bi- and uni-directional causal nonlinear relationships in the return series. Finally, we investigate the hypothesis of nonlinear non-causality after controlling for conditional heteroskedasticity in the data using a GARCH-BEKK model. Our approach allows the entire variance-covariance structure of the currency interrelationship to be incorporated in order to explicitly capture the volatility spillover mechanism. Whilst the nonparametric test statistics are smaller in some cases, significant nonlinear causal linkages persisted even after GARCH filtering during both the pre- and post-Asian crisis period. This indicates that currency returns may exhibit asymmetries and statistically significant higher-order moments.

A new statistic and practical guidelines for nonparametric Granger causality testing

Upon illustrating how smoothing may cause over-rejection in nonparametric tests for Granger non-causality, we propose a new test statistic for which problems of this type can be avoided. We develop asymptotic theory for the new test statistic, and perform a simulation study to investigate the properties of the new test in comparison with its natural counterpart, the Hiemstra-Jones test. Our simulation results indicate that, if the bandwidth tends to zero at the appropriate rate as the sample size increases, the size of the new test remains close to nominal, while the power remains large. Transforming the time series to uniform marginals improves the behavior of both tests. In applications to Standard and Poor's index volumes and returns, the Hiemstra-Jones test suggests that volume Granger-causes returns. However, the evidence for this gets weaker if we carefully apply the recommendations suggested by our simulation study.

The Relationship between Crude Oil Spot and Futures Prices: Cointegration, Linear and Nonlinear Causality

The present study investigates the linear and nonlinear causal linkages between daily spot and futures prices for maturities of one, two, three and four months of West Texas Intermediate (WTI) crude oil. The data cover two periods October 1991-October 1999 and November 1999-October 2007, with the latter being significantly more turbulent. Apart from the conventional linear Granger test we apply a new nonparametric test for nonlinear causality by Diks and Panchenko after controlling for cointegration. In addition to the traditional pairwise analysis, we test for causality while correcting for the effects of the other variables. To check if any of the observed causality is strictly nonlinear in nature, we also examine the nonlinear causal relationships of VECM filtered residuals. Finally, we investigate the hypothesis of nonlinear non-causality after controlling for conditional heteroskedasticity in the data using a GARCH-BEKK model. Whilst the linear causal relationships disappear after VECM cointegration filtering, nonlinear causal linkages in some cases persist even after GARCH filtering in both periods. This indicates that spot and futures returns may exhibit asymmetries and statistically significant higher-order moments. Moreover, the results imply that if nonlinear effects are accounted for, neither market leads or lags the other consistently, videlicet the pattern of leads and lags changes over time.

Nonparametric Tests for Serial Independence Based on Quadratic Forms

Tests for serial independence and goodness-of-fit based on divergence notions between probability distributions, such as the Kullback-Leibler divergence or Hellinger distance, have recently received much interest in time series analysis. The aim of this paper is to introduce tests for serial independence using kernel-based quadratic forms. This separates the problem of consistently estimating the divergence measure from that of consistently estimating the underlying joint densities, the existence of which is no longer required. Exact level tests are obtained by implementing a Monte Carlo procedure using permutations of the original observations. The bandwidth selection problem is addressed by introducing a multiple bandwidth procedure based on a range of different bandwidth values. After numerically establishing that the tests perform well compared to existing nonparametric tests, applications to estimated time series residuals are considered. The approach is illustrated with an application to financial returns data.

Equivalence and bifurcations of finite order stochastic processes

This article presents an equivalence notion of finite order stochastic processes. Local dependence measures are defined in terms of joint and marginal densities. The dependence measures are classified topologically using level sets. The corresponding bifurcation theory is illustrated with some simple examples.

Rank-based entropy tests for serial independence

In nonparametric tests for serial independence the marginal distribution of the data acts as an infinite dimensional nuisance parameter. The decomposition of joint distributions in terms of a copula density and marginal densities shows that in general empirical marginals carry no information on dependence. It follows that the order of ranks is sufficient for inference, which motivates transforming the data to a pre-specified marginal distribution prior to testing. As a test statistic we use an estimator of the marginal redundancy, which has some desirable properties in the case of transforming to uniform marginals. We numerically study the finite sample properties of these tests when the data are transformed to uniform as well as normal marginals. The performance of the tests is compared with that of the BDS test as well as with a parametric rank-based test against ARCH alternatives.