Long-range Dependence Parameter Research Articles

Comparison of standard long memory time series

Standard long memory models are in abundance in the literature today. Selecting the best such a model in terms of capturing key requisite features and trends in data becomes a challenge. This paper addresses the issue through a sequence of Monte Carlo experiments on simulated data and introduces an interval estimate on the asymptotic variance for the long-range dependence parameter of the entire family of standard long memory time series considered within the scope of the study.

Informational assessment of large scale self-similarity in nonlinear random field models

Large-scale behavior of a wide class of spatial and spatiotemporal processes is characterized in terms of informational measures. Specifically, subordinated random fields defined by nonlinear transformations on the family of homogeneous and isotropic Lancaster–Sarmanov random fields are studied under long-range dependence (LRD) assumptions. In the spatial case, it is shown that Shannon mutual information between random field components for infinitely increasing distance, which can be properly interpreted as a measure of large scale structural complexity and diversity, has an asymptotic power law decay that depends on the underlying LRD parameter scaled by the subordinating function rank. Sensitivity with respect to distortion induced by the deformation parameter under the generalized form given by divergence-based Rényi mutual information is also analyzed. In the spatiotemporal framework, a spatial infinite-dimensional random field approach is adopted. The study of the large-scale asymptotic behavior is then extended under the proposal of a functional formulation of the Lancaster–Sarmanov random field class, as well as of divergence-based mutual information. Results are illustrated, in the context of geometrical analysis of sample paths, considering some scenarios based on Gaussian and Chi-Square subordinated spatial and spatiotemporal random fields.

Measuring price efficiency in petroleum markets: New insights using various long-range dependence techniques

This paper empirically measures pricing efficiency in the petroleum markets using a battery of long-range dependence techniques. We analyse the volatility (absolute returns) of fourteen petroleum products using daily prices from January 1990 to April 2020. We use bootstrapping techniques to estimate the confidence intervals of the long-range dependence parameters. Overall results obtained using CMA-MAF, CMA-MSF, GPH and periodogram techniques suggest that there is no evidence of long-range dependence in absolute returns of petroleum markets prices indices at least at the 10% level of significance. For robustness purposes, findings from the rolling windows estimates reveal that the results are not affected by the normality assumption under different rolling window sizes. Furthermore, we find that Gas oil, Biofuel and Heating oil are the most correctly priced while Crude Oil Dubai emerging is the least efficient. Comparing the efficiency index of the all six crude oil returns in our sample, WTI oil was seen to be less efficient than Brent crude oil. In all, our work uncovers crucial implications for investors and policymakers.

Asymptotic normality of wavelet covariances and multivariate wavelet Whittle estimators

Multivariate processes with long-range dependence properties can be encountered in many fields of application. Two fundamental characteristics in such frameworks are long-range dependence parameters and correlations between component time series. We consider multivariate long-range dependent linear processes, not necessarily Gaussian. We show that the covariances between the wavelet coefficients in this setting are asymptotically Gaussian. We also study the asymptotic distributions of the estimators of the long-range dependence parameter and the long-run covariance by a wavelet-based Whittle procedure. We prove the asymptotic normality of the estimators, and we provide an explicit expression for the asymptotic covariances. An empirical illustration of this result is proposed on a real dataset of rat brain connectivity.

Investigating Long-Range Dependence of Emerging Asian Stock Markets Using Multifractal Detrended Fluctuation Analysis

The use of multifractal approaches has been growing because of the capacity of these tools to analyze complex properties and possible nonlinear structures such as those in financial time series. This paper analyzes the presence of long-range dependence and multifractal parameters in the stock indices of nine MSCI emerging Asian economies. Multifractal Detrended Fluctuation Analysis (MFDFA) is used, with prior application of the Seasonal and Trend Decomposition using the Loess (STL) method for more reliable results, as STL separates different components of the time series and removes seasonal oscillations. We find a varying degree of multifractality in all the markets considered, implying that they exhibit long-range correlations, which could be related to verification of the fractal market hypothesis. The evidence of multifractality reveals symmetry in the variation trends of the multifractal spectrum parameters of financial time series, which could be useful to develop portfolio management. Based on the degree of multifractality, the Chinese and South Korean markets exhibit the least long-range dependence, followed by Pakistan, Indonesia, and Thailand. On the contrary, the Indian and Malaysian stock markets are found to have the highest level of dependence. This evidence could be related to possible market inefficiencies, implying the possibility of institutional investors using active trading strategies in order to make their portfolios more profitable.

Estimation of long-range dependence in gappy Gaussian time series.

Knowledge of the long range dependence (LRD) parameter is critical to studies of self-similar behavior. However, statistical estimation of the LRD parameter becomes difficult when the observed data are masked by short range dependence and other noise, or are gappy in nature (i.e., some values are missing in an otherwise regular sampling). Currently there is a lack of theory for spectral- and wavelet-based estimators of the LRD parameter for gappy data. To address this, we estimate the LRD parameter for gappy Gaussian semiparametric time series based upon undecimated wavelet variances. We develop estimation methods by using novel estimators of the wavelet variances, providing asymptotic theory for the joint distribution of the wavelet variances and our estimator of the LRD parameter. We introduce sandwich estimators to compute standard errors for our estimates. We demonstrate the efficacy of our methods using Monte Carlo simulations, and provide guidance on practical issues such as how to select the range of wavelet scales. We demonstrate the methodology using two applications: one for gappy Arctic sea-ice draft data, and another for gap free and gappy daily average temperature data collected at 17 locations in south central Sweden.

Directional dependency and coastal framework geology: implications for barrier island resilience

Abstract. Barrier island transgression is influenced by the alongshore variation in beach and dune morphology, which determines the amount of sediment moved landward through wash-over. While several studies have demonstrated how variations in dune morphology affect island response to storms, the reasons for that variation and the implications for island management remain unclear. This paper builds on previous research by demonstrating that paleo-channels in the irregular framework geology can have a directional influence on alongshore beach and dune morphology. The influence of relict paleo-channels on beach and dune morphology on Padre Island National Seashore, Texas, was quantified by isolating the long-range dependence (LRD) parameter in autoregressive fractionally integrated moving average (ARFIMA) models, originally developed for stock market economic forecasting. ARFIMA models were fit across ∼250 unique spatial scales and a moving window approach was used to examine how LRD varied with computational scale and location along the island. The resulting LRD matrices were plotted by latitude to place the results in the context of previously identified variations in the framework geology. Results indicate that the LRD is not constant alongshore for all surface morphometrics. Many flares in the LRD plots correlate to relict infilled paleo-channels, indicating that the framework geology has a significant influence on the morphology of Padre Island National Seashore (PAIS). Barrier island surface morphology LRD is strongest at large paleo-channels and decreases to the north. The spatial patterns in LRD surface morphometrics and framework geology variations demonstrate that the influence of paleo-channels can be asymmetric (i.e., affecting beach–dune morphology preferentially in one direction alongshore) where the alongshore sediment transport gradient was unidirectional during island development. The asymmetric influence of framework geology on coastal morphology has long-term implications for coastal management activities because it dictates the long-term behavior of a barrier island. Coastal management projects should first seek to assess the framework geology and understand how it influences coastal processes in order to more effectively balance long-term natural variability with short-term societal pressure.

Systematic inference of the long-range dependence and heavy-tail distribution parameters of ARFIMA models

Long-Range Dependence (LRD) and heavy-tailed distributions are ubiquitous in natural and socio-economic data. Such data can be self-similar whereby both LRD and heavy-tailed distributions contribute to the self-similarity as measured by the Hurst exponent. Some methods widely used in the physical sciences separately estimate these two parameters, which can lead to estimation bias. Those which do simultaneous estimation are based on frequentist methods such as Whittle’s approximate maximum likelihood estimator. Here we present a new and systematic Bayesian framework for the simultaneous inference of the LRD and heavy-tailed distribution parameters of a parametric ARFIMA model with non-Gaussian innovations. As innovations we use the α-stable and t-distributions which have power law tails. Our algorithm also provides parameter uncertainty estimates. We test our algorithm using synthetic data, and also data from the Geostationary Operational Environmental Satellite system (GOES) solar X-ray time series. These tests show that our algorithm is able to accurately and robustly estimate the LRD and heavy-tailed distribution parameters.

Detecting long-range dependence in non-stationary time series

An important problem in time series analysis is the discrimination between non-stationarity and longrange dependence. Most of the literature considers the problem of testing specic parametric hypotheses of non-stationarity (such as a change in the mean) against long-range dependent stationary alternatives. In this paper we suggest a simple nonparametric approach, which can be used to test the null-hypothesis of a general non-stationary short-memory against the alternative of a non-stationary long-memory process. This test is working in the spectral domain and uses a sieve of approximating tvFARIMA models to estimate the time varying long-range dependence parameter nonparametrically. We prove uniform consistency of this estimate and asymptotic normality of an averaged version. These results yield a simple test (based on the quantiles of the standard normal distribution), and it is demonstrated in a simulation study that - despite of its nonparametric nature - the new test outperforms the currently available methods, which are constructed to discriminate between specic parametric hypotheses of non-stationarity short- and stationarity long-range dependence.

Estimation methods for the LRD parameter under a change in the mean

When analyzing time series which are supposed to exhibit long-range dependence (LRD), a basic issue is the estimation of the LRD parameter, for example the Hurst parameter $$H \in (1/2, 1)$$ . Conventional estimators of H easily lead to spurious detection of long memory if the time series includes a shift in the mean. This defect has fatal consequences in change-point problems: Tests for a level shift rely on H, which needs to be estimated before, but this estimation is distorted by the level shift. We investigate two blocks approaches to adapt estimators of H to the case that the time series includes a jump and compare them with other natural techniques as well as with estimators based on the trimming idea via simulations. These techniques improve the estimation of H if there is indeed a change in the mean. In the absence of such a change, the methods little affect the usual estimation. As adaption, we recommend an overlapping blocks approach: If one uses a consistent estimator, the adaption will preserve this property and it performs well in simulations.

A wavelet lifting approach to long-memory estimation

Reliable estimation of long-range dependence parameters is vital in time series. For example, in environmental and climate science such estimation is often key to understanding climate dynamics, variability and often prediction. The challenge of data collection in such disciplines means that, in practice, the sampling pattern is either irregular or blighted by missing observations. Unfortunately, virtually all existing Hurst parameter estimation methods assume regularly sampled time series and require modification to cope with irregularity or missing data. However, such interventions come at the price of inducing higher estimator bias and variation, often worryingly ignored. This article proposes a new Hurst exponent estimation method which naturally copes with data sampling irregularity. The new method is based on a multiscale lifting transform exploiting its ability to produce wavelet-like coefficients on irregular data and, simultaneously, to effect a necessary powerful decorrelation of those coefficients. Simulations show that our method is accurate and effective, performing well against competitors even in regular data settings. Armed with this evidence our method sheds new light on long-memory intensity results in environmental and climate science applications, sometimes suggesting that different scientific conclusions may need to be drawn.

Fractional bivariate exponential estimator for long-range dependent random field

Estimating long-range dependence parameter of a random field, which provides a measure for the extent of long-range dependence, is a challenging problem. Fractional Bivariate EXponential (FBEXP) estimator is proposed for long-range dependence parameter estimation in random fields observed on a regular lattice. FBEXP is a generalized version of the commonly used FEXP estimator in time series. FBEXP estimator belongs to the class of broadband semiparametric estimators, which makes it free from the optimal bandwidth selection problem, present in other semiparametric estimators. The behavior of FBEXP estimator depends on the model order of bivariate exponential model. Three data-driven model order selection criteria are introduced to serve as a guide in appropriate choice of model order for efficient estimation of long-range dependence using FBEXP. The finite sample performance of the FBEXP estimator and model order selection criteria are illustrated via simulation study. FBEXP estimator provides an efficient estimate for the long-range dependence parameter, especially when the spectral density is sufficiently smooth everywhere except at the points of singularities. Total column ozone amounts in Europe and Mediterranean region illustrate the applicability of FBEXP estimator in providing efficient estimates for direction specific long-range dependence parameters and the spectral density of the process.

On a class of minimum contrast estimators for Gegenbauer random fields

The article introduces spatial long-range dependent models based on the fractional difference operators associated with the Gegenbauer polynomials. The results on consistency and asymptotic normality of a class of minimum contrast estimators of long-range dependence parameters of the models are obtained. A methodology to verify assumptions for consistency and asymptotic normality of minimum contrast estimators is developed. Numerical results are presented to confirm the theoretical findings.

Large scale reduction principle and application to hypothesis testing

Consider a non–linear function $G(X_{t})$ where $X_{t}$ is a stationary Gaussian sequence with long–range dependence. The usual reduction principle states that the partial sums of $G(X_{t})$ behave asymptotically like the partial sums of the first term in the expansion of $G$ in Hermite polynomials. In the context of the wavelet estimation of the long–range dependence parameter, one replaces the partial sums of $G(X_{t})$ by the wavelet scalogram, namely the partial sum of squares of the wavelet coefficients. Is there a reduction principle in the wavelet setting, namely is the asymptotic behavior of the scalogram for $G(X_{t})$ the same as that for the first term in the expansion of $G$ in Hermite polynomial? The answer is negative in general. This paper provides a minimal growth condition on the scales of the wavelet coefficients which ensures that the reduction principle also holds for the scalogram. The results are applied to testing the hypothesis that the long-range dependence parameter takes a specific value.

Fractionally Integrated Panel Data Systems

We consider large n, T panel data models with fixed effects, persistent common factors allowing for cross-section dependence, and possibly correlated persistent innovations that are assumed fractionally integrated. In the model, a) persistence in the innovations and the common factor allows for cointegrating relationships in the unobserved idiosyncratic components; b) the vector of innovations, whose elements can be contemporaneously correlated, can also exhibit short-memory dynamics in the form of a finite-order vector autoregressive process; c) deterministic linear and nonlinear trends can be nested. We perform estimations on the defactored observed series where the projections are based on the sample averages of fractionally differenced data. In the estimation, we use a computationally convenient equation-by-equation conditional-sum-of-squares (CSS) criterion, leading to GLS-type estimates for slope parameters. The CSS estimates of individual slope and long-range dependence parameters are root-T consistent while the mean-group slope estimate is root-n consistent, all with centered asymptotic normal distributions, irrespective of cointegration. A study of small-sample performance is carried out showing desirable properties for our estimation method. Finally, an empirical application to the long-run relationship between real GDP and debt is included.

On distinguishing multiple changes in mean and long-range dependence using local Whittle estimation

It is well known that changes in mean superimposed by a short-range dependent series can be confused easily with long-range dependence. A procedure to distinguish the two phenomena is introduced. The proposed procedure is based on the local Whittle estimation of the long-range dependence parameter applied to the series after removing changes in mean, and comparing the results to those obtained through the available CUSUM-like approaches. According to the proposed procedure, for example, volatility series in finance seem more consistent with the changes-in-mean models whereas hydrology and telecommunication series are more in line with long-range dependence. As part of this work, a new method based on the local Whittle estimation to find the number of breaks is also introduced and its consistency is proved for the changes-in-mean models.

Time-varying long-range dependence in stock market returns and financial market disruptions – a case of eight European countries

The long-range dependence (or long memory) in stock market returns has many implications for modern financial economics. The existent empirical studies on long-range dependence in stock market returns, however, do not examine it on a dynamical basis. In this article we applied a rolling window approach to prove that long-range dependence parameter for eight European stock market returns is time-varying. Our findings show that sharp, but temporary, increases of long-range dependence parameter for investigated stock market returns in the period October 1999 to April 2011 coincided with the major financial market disruptions in the world and Europe.

Bootstrap Estimation of Long-Range Dependence in Arfima Processes

Long-memory or long-range dependence (LRD) denotes the property of a time series to exhibit a significant dependence between very distant observations. The presence of long-memory is established in diverse fields, for example, in hydrology, finance, network traffic, psychology, etc. From a practical point of view an important issue is the estimation of LRD parameters, especially the estimation of the Hurst parameter proposed in [19]. The Hurst parameter, usually referred as the H parameter, indicates the intensity of LRD.Various methods for estimating the H parameter in a time series data are available some of which are described in [3]. They are validated by appealing to an asymptotic analysis supposing that the sample size of the time series converges to infinity. Taqqu et al. [37] investigated empirical performance of nine methods for estimating the LRD in simulated autoregressive fractionally integrated moving average (ARFIMA) processes for a fairly large sample size of 10000 terms.The bootstrap is a general statistical procedure for statistical inference, originally introduced by Efron [11] that generally yields better estimations in finite sample sizes, than the classical methods. In this paper, we consider a bootstrap technique, which uses blocks composed of cycles, to estimate the H parameter in ARFIMA processes. The bias and the root mean square error (RMSE) of five estimators and their corresponded bootstrap estimators are obtained by a Monte Carlo study.

α-stable laws for noncoding regions in DNA sequences

In this work, we analyze the long-range dependence parameter for a nucleotide sequence in several different transformations. The long-range dependence parameter is estimated by the approximated maximum likelihood method, by a novel estimator based on the spectral envelope theory, by a regression method based on the periodogram function, and also by the detrended fluctuation analysis method. We study the length distribution of coding and noncoding regions for all Homo sapiens chromosomes available from the European Bioinformatics Institute. The parameter of the tail rate decay is estimated by the Hill estimator ˆα. We show that the tail rate decay is greater than 2 for coding regions, while for almost all noncoding regions it is less than 2.

Characterization of long-range dependent traffic regulated by leaky-bucket policers and shapers

Long-range dependence (LRD) is a widely verified property of Internet traffic, which severely affects network queuing performance. An approach for guaranteeing quality-of-service requirements is enforcing the statistical profile of input traffic by policing or shaping regulators. In this paper, it is investigated by thorough simulation how leaky-bucket (LB) policers and shapers affect the LRD of regulated traffic having 1 / f α power spectral density. Unlike previous studies on this subject, spectral analysis of regulated traffic and estimation of its LRD parameter α are carried out in the time domain using the Modified Allan Variance, because of its demonstrated superior accuracy in LRD parameter estimation. Adoption of this tool allows to attain unprecedented precision and fineness in characterizing the LRD of LB regulated traffic. In addition, the probability that LRD traffic is dropped by policers or that it exceeds a delay limit in shapers is also studied. The queuing behaviour of LRD regulated traffic in FIFO schedulers is finally investigated, highlighting conditions under which a service level agreement based on delay bounds can be violated by varying α in input LRD traffic, even if this is controlled by policers or shapers.