Testing for long-range dependence in the presence of shifting means or a slowly declining trend, using a variance-type estimator

In this paper we examine the effects of certain types of non- stationarity on the detection of long-range dependence and on the estimation of the Hurst parameter H , when using a variance-type estimator. The resulting estimate of H can be misleading when the series has either a jump in the mean or a slow trend. In such a case, plotting the logarithm of the variance versus the logarithm of the level of aggregation gives a curve which is quite different from a straight line. A method for distinguishing between the effects of long-range dependence and these types of non-stationarity is developed.     

On the asymptotic properties of a feasible estimator of the continuous time long memory parameter

This article considers a fractional noise model in continuous time and examines the asymptotic properties of a feasible frequency domain maximum likelihood estimator of the long memory parameter. The feasible estimator is one that maximizes an approximation to the likelihood function (the approximation arises from the fact that the spectral density function involves the finite truncation of an infinite summation). It is of interest therefore to explore the conditions required of this approximation to ensure the consistency and asymptotic normality of this estimator.     

Temporal Aggregation and Bandwidth selection in estimating long memory

This article aims at showing that a temporal aggregation and a specific bandwidth reduction lead to the same asymptotic properties in estimating long memory by Geweke and Porter‐Hudak's [Journal of Time Series Analysis (1983 ) vol. 4, pp. 221–237] and Robinson's [Annals of Statistics (1995b ) vol. 23, pp. 1630–1661] estimators. In other words, irrespective of the level of temporal aggregation, the asymptotic properties of the estimator are uniquely determined by the number of periodogram ordinates used in the estimation, provided some mild additional assumptions are imposed. Monte Carlo simulations show that this result is a good approximation in finite samples. A real example with the daily US Dollar/French Franc exchange rate series is also provided.     

Testing for a break in persistence under long-range dependencies

Abstract.  We show that tests for a break in the persistence of a time series in the classical I (0)/ I (1) framework have serious size distortions when the actual data-generating process (DGP) exhibits long-range dependencies. We prove that the limiting distribution of a CUSUM of squares-based test depends on the true memory parameter if the DGP exhibits long memory. We propose adjusted critical values for the test and give finite sample response curves that allow easy implementation of the test by the practitioner and also ease in computing the relevant critical values. We furthermore prove the consistency of the test for a simple breakpoint estimator also under long memory. We show that the test has satisfying power properties when the correct critical values are used.     

An optimal stopping time for a power law process

Exact reliability analysis for complex repairable systems are usually difficult because of the complicated failure process that can result from repair policy. A common approach in practice is to use a simplified process such as the Power Law process which, although not exact, yields useful practical results. In this paper we consider the problem of determining a stopping time when estimating parameters of a Power Law process. The Bayes method is used in order to obtain an optimal time.     

Using Difference‐Based Methods for Inference in Regression with Fractionally Integrated Processes

Abstract. This paper suggests a difference‐based method for inference in the regression model involving fractionally integrated processes. Under suitable regularity conditions, our method can effectively deal with the inference problems associated with the regression model consisting of nonstationary, stationary and intermediate memory regressors, simultaneously. Although the difference‐based method provides a very flexible modelling framework for empirical studies, the implementation of this method is extremely easy, because it completely avoids the difficult problems of choosing a kernel function, a bandwidth parameter, or an autoregressive lag length for the long‐run variance estimation. The asymptotic local power of our method is investigated with a sequence of local data‐generating processes (DGP) in what Davidson and MacKinnon [Canadian Journal of Economics. (1985) Vol. 18, pp. 38–57] call ‘regression direction’. The simulation results indicate that the size control of our method is excellent even when the sample size is only 100, and the pattern of power performance is highly consistent with the theoretical finding from the asymptotic local power analysis conducted in this paper.     

The exact discrete model of a system of linear stochastic differential equations driven by fractional noise

Abstract. This paper derives the exact discrete model (EDM) of a kth‐order system of stochastic differential equations driven by a vector fractional noise under fixed initial conditions. The EDM can be used for the Gaussian estimation and forecasting with long‐memory discrete‐time equispaced data. Detailed formulae which are necessary for the construction and numerical evaluation of the Gaussian likelihood under two observation schemes are established. State variables can be observed either at equispaced points in time or as integrals over the observational interval.     

Throughput-pressure relationships for power law fluids in single screw extruders

A simple expression is presented describing the volumetric throughput in a single screw extruder as a function of the pressure gradient with the dependence on the power law index included. Analytical expressions have also been developed for the optimum channel depth and helix angle for power law fluids.     

A joint test of fractional integration and structural breaks at a known period of time

Abstract.  We propose the use of a version of the tests of Robinson [Journal of the American Statistical Association, 89 (1994) 1420] for testing the order of integration in raw time series in the presence of structural breaks at known periods of time. Also, a joint test for simultaneously testing the degree of integration and the need for the break is developed. Several Monte Carlo experiments conducted in the paper show that the joint test has better size and power properties relative to Robinson's (1994) tests. The tests are applied to the annual structure of the US real GDP, the results showing that the series is I ( d ) with d ≥1, and with a break due to the World War II.     

Bayesian methods for change-point detection in long-range dependent processes

Abstract. We describe a Bayesian method for detecting structural changes in a long-range dependent process. In particular, we focus on changes in the long-range dependence parameter, d , and changes in the process level, μ . Markov chain Monte Carlo (MCMC) methods are used to estimate the posterior probability and size of a change at time t , along with other model parameters. A time-dependent Kalman filter approach is used to evaluate the likelihood of the fractionally integrated ARMA model characterizing the long-range dependence. The method allows for multiple change points and can be extended to the long-memory stochastic volatility case. We apply the method to three examples, to investigate a change in persistence of the yearly Nile River minima, to investigate structural changes in the series of durations between intraday trades of IBM stock on the New York Stock Exchange, and to detect structural breaks in daily stock returns for the Coca Cola Company during the 1990s.     

Estimating a change point in the long memory parameter

We propose an estimator of change point in the long memory parameter d of an ARFIMA(p, d, q) process using the sup Wald test. We derive the consistency and the rate of convergence of the estimator for the time of change. The convergence rate of our change point estimator depends on the magnitude of a shift. Furthermore, we obtain the limiting distribution of our change point estimator without depending on the distribution of the process. Therefore, we can construct confidence intervals for the change point. Simulations show the validity of the asymptotic theory of our estimator if the sample size is large enough. We apply our change point estimator to the yearly Nile river minimum water level.     

Hindered settling in non-newtonian power law liquids

New experimental results on the hindered settling of model glass bead suspensions in non-Newtonian suspending media are reported. The data presented encompass the following ranges of variables: 7.38 × 10^?4 ≤ Re_1∞ ≤ 2; 0.0083 ≤ d/D ≤ 0.0703; 0.13 ≤ C ≤ 0.43 and 1 ≥ n ≥ 0.8. In these ranges of conditions, the dependence of the hindered settling velocity on concentration is adequately represented by the corresponding Newtonian expressions available in the literature. The influence of the power law flow behaviour index is completely embodied in the modified definition of the Reynolds number used for power law liquids.     

Settling of cylinders in power law liquids

New measurements on drag coefficient and wall effects on the free settling motion of cylinders in two shearthinning polymer solutions are reported. The ranges of conditions covered in this study are: 1 < Re_∞∞ < 40; 0.25 < (LID) < 2; 0.079 < dID < 0.4, and n = 0.65 and 0.74. Both wall correction factor and drag coefficient results are in line with the Newtonian behavior provided a modified Reynolds number is used to represent the results.     

Semiparametric Inference in Seasonal and Cyclical Long Memory Processes

Several semiparametric estimates of the memory parameter in standard long memory time series are now available. They consider only local behaviour of the spectrum near zero frequency, about which the spectrum is symmetric. However long-range dependence can appear as a spectral pole at any Nyqvist frequency (reflecting seasonal or cyclical long-memory), where the spectrum need display no such symmetry. We introduce Seasonal/Cyclical Asymmetric Long Memory (SCALM) processes that allow differing rates of increase on either side of such a pole. To estimate the two consequent memory parameters we extend two semiparametric methods that were proposed for the standard case of a spectrum diverging at the origin, namely the log-periodogram and Gaussian or Whittle methods. We also provide three tests of symmetry. Monte Carlo analysis of finite sample behaviour and an empirical application to UK inflation data are included. Our models and methods allow also for the possibility of negative dependence, described by a possibly asymmetric spectral zero.     

On the properties of the periodogram of a stationary long‐memory process over different epochs with applications

This article studies the asymptotic properties of the discrete Fourier transforms (DFT) and the periodogram of a stationary long‐memory time series over different epochs. The main theoretical result is a novel bound for the covariance of the DFT ordinates evaluated on two distinct epochs, which depends explicitly on the Fourier frequencies and the gap between the epochs. This result is then applied to obtain the limiting distribution of some nonlinear functions of the periodogram over different epochs, under the additional assumption of gaussianity. We then apply this result to construct an estimator of the memory parameter based on the regression in a neighbourhood of the zero‐frequency of the logarithm of the averaged periodogram, obtained by computing the empirical mean of the periodogram over adjacent epochs. It is shown that replacing the periodogram by its average has an effect similar to the frequency domain pooling to reduce the variance of the estimate. We also propose a simple procedure to test the stationarity of the memory coefficient. A limited Monte Carlo experiment is presented to support our findings.     

Rising velocity of a swarm of spherical bubbles in a power law non-newtonian liquid

Previous work on the rise of a swarm of bubbles in non-Newtonian media has been reviewed. Variational principles have been combined with the Happel free surface cell model to obtain upper and lower bounds on the swarm velocities of bubbles rising slowly through power law liquids. The predictions presented herein encompass wide ranges of gas holdup and shear-thinning behaviour.     

Predictions of heat transfer and pressure drop for a modified power law fluid flow in a square duct

Numerical Solutions for the Nusselt Numbers (CHF and CWT) and the Friction Factor times Reynolds Number have been obtained for fully developed laminar flow of a MPL (Modified Power Law) fluid within a square duct. The solutions are applicable to pseudoplastic fluids over a wide shear rate range from Newtonian at low shear rates through a transition region to power law behavior at higher shear rates. A shear rate parameter is identified, which allows the prediction of the shear rate range for a specified set of operating conditions. Numerical results of the Nusselt numbers (CHF and CWT) and the Friction factors times Reynolds number for the Newtonian and power law regions are compared with previous published results, showing agreement with 0.02% in Newtonian region and 4.0% in power law region.