1,304 research outputs found

    Partial period autocorrelations of geometric sequences

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    Nonstationary Nonlinearity: An Outlook for New Opportunities

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    In this paper, we look for new opportunities that can be exploited using some of the recent developments on the theory of nonlinear models with integrated time series. Heuristic introductions on the basic tools and asymptotics are followed by the opportunities in three different directions: in data generation, in mean and in volatility. In the direction of data generation, we investigate the nonlinear transformations of random walks. It is shown in particular that they can generate stationary long memory as well as bounded nonstationarity and leptokurticity, which we commonly observe in many of economic and financial data. We then discuss how the nonlinear mean relationships between integrated processes can be appropriately formulated, interpreted and estimated within the regression framework. Both the nonlinear least squares regression and the nonparametric kernel regression are considered. Such formulations allow us to explore the nonlinear and nonparametric cointegration, which may be used in modelling the nonlinear and nonparametric longrun relationships among various economic and financial time series. Finally, a stochastic volatility model with the conditional variance specified as a nonlnear function of a random walk is examined. Established are various time series properties of the model, which are shown to be largely consistent with the observed characteristics of many time series data.

    Binary sequences with prescribed autocorrelations

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    The Chinese Chaos Game

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    The yuan-dollar returns prior to the 2005 revaluation show a Sierpinski triangle in an iterated function system clumpiness test. Yet the fractal vanishes after the revaluation. The Sierpinski commonly emerges in the chaos game, where randomness coexists with deterministic rules [2, 3]. Here it is explained by the yuan’s pegs to the US dollar, which made more than half of the data points close to zero. Extra data from the Brazilian and Argentine experiences do confirm that the fractal emerges whenever exchange rate pegs are kept for too long.

    Integer-valued time series and renewal processes

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    This research proposes a new but simple model for stationary time series of integer counts. Previous work in the area has focused on mixture and thinning methods and links to classical time series autoregressive moving-average difference equations; in contrast, our methods use a renewal process to generate a correlated sequence of Bernoulli trials. By superpositioning independent copies of such processes, stationary series with binomial, Poisson, geometric, or any other discrete marginal distribution can be readily constructed. The model class proposed is parsimonious, non-Markov, and readily generates series with either short or long memory autocovariances. The model can be fitted with linear prediction techniques for stationary series. Estimation of process parameters based on conditional least squares methods is considered. Asymptotic properties of the estimators are derived. The models sometimes have an autoregressive moving-average structure and we consider the AR(1) count process case in detail. Unlike previous methods based on mixture and thinning tactics, series with negative autocorrelations can be produced

    Time series models for discrete data

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    Cost of the Generalised Hybrid Monte Carlo Algorithm for Free Field Theory

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    We study analytically the computational cost of the Generalised Hybrid Monte Carlo (GHMC) algorithm for free field theory. We calculate the Metropolis acceptance probability for leapfrog and higher-order discretisations of the Molecular Dynamics (MD) equations of motion. We show how to calculate autocorrelation functions of arbitrary polynomial operators, and use these to optimise the GHMC momentum mixing angle, the trajectory length, and the integration stepsize for the special cases of linear and quadratic operators. We show that long trajectories are optimal for GHMC, and that standard HMC is more efficient than algorithms based on Second Order Langevin Monte Carlo (L2MC), sometimes known as Kramers Equation. We show that contrary to naive expectations HMC and L2MC have the same volume dependence, but their dynamical critical exponents are z = 1 and z = 3/2 respectively.Comment: 54 pages, 3 figure
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