Long-range Dependent Processes Research Articles

Editorial: Long-range dependent processes: theory and applications, volume II

Editorial: Long-range dependent processes: theory and applications, volume II

Joint Sum-and-Max Limit for a Class of Long-Range Dependent Processes with Heavy Tails

Joint Sum-and-Max Limit for a Class of Long-Range Dependent Processes with Heavy Tails

On optimal block resampling for Gaussian-subordinated long-range dependent processes

Block-based resampling estimators have been intensively investigated for weakly dependent time processes, which has helped to inform implementation (e.g., best block sizes). However, little is known about resampling performance and block sizes under strong or long-range dependence. To establish guideposts in block selection, we consider a broad class of strongly dependent time processes, formed by a transformation of a stationary long-memory Gaussian series, and examine block-based resampling estimators for the variance of the prototypical sample mean; extensions to general statistical functionals are also considered. Unlike weak dependence, the properties of resampling estimators under strong dependence are shown to depend intricately on the nature of nonlinearity in the time series (beyond Hermite ranks) in addition to the long-memory coefficient and block size. Additionally, the intuition has often been that optimal block sizes should be larger under strong dependence (say O(n1/2) for a sample size n) than the optimal order O(n1/3) known under weak dependence. This intuition turns out to be largely incorrect, though a block order O(n1/2) may be reasonable (and even optimal) in many cases, owing to nonlinearity in a long-memory time series. While optimal block sizes are more complex under long-range dependence compared to short-range, we provide a consistent data-driven rule for block selection. Numerical studies illustrate that the guides for block selection perform well in other block-based problems with long-memory time series, such as distribution estimation and strategies for testing Hermite rank.

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.

On almost sure limit theorems for heavy-tailed products of long-range dependent linear processes

Marcinkiewicz strong law of large numbers, n−1p∑k=1n(dk−d)→0 almost surely with p∈(1,2), are developed for products dk=∏r=1sxk(r), where xk(r)=∑l=−∞∞ck−l(r)ξl(r) are two-sided linear processes with coefficients {cl(r)}l∈Z and i.i.d. zero-mean innovations {ξl(r)}l∈Z. The decay of the coefficients cl(r) as |l|→∞, can be slow enough for {xk(r)} to have long memory while {dk} can have heavy tails. The long-range dependence and heavy tails for {dk} are handled simultaneously and a decoupling property shows the convergence rate is dictated by the worst of long-range dependence and heavy tails, but not their combination. The Marcinkiewicz strong law of large numbers is also extended to the multivariate linear process case.

Generalized Cauchy Process: Difference Iterative Forecasting Model

The contribution of this article is mainly to develop a new stochastic sequence forecasting model, which is also called the difference iterative forecasting model based on the Generalized Cauchy (GC) process. The GC process is a Long-Range Dependent (LRD) process described by two independent parameters: Hurst parameter H and fractal dimension D. Compared with the fractional Brownian motion (fBm) with a linear relationship between H and D, the GC process can more flexibly describe various LRD processes. Before building the forecasting model, this article demonstrates the GC process using H and D to describe the LRD and fractal properties of stochastic sequences, respectively. The GC process is taken as the diffusion term to establish a differential iterative forecasting model, where the incremental distribution of the GC process is obtained by statistics. The parameters of the forecasting model are estimated by the box dimension, the rescaled range, and the maximum likelihood methods. Finally, a real wind speed data set is used to verify the performance of the GC difference iterative forecasting model.

Robust wavelet-based estimation for varying coefficient dynamic models under long-dependent structures

This paper considers a class of robust estimation problems for varying coefficient dynamic models via wavelet techniques, which can adapt to local features of the underlying functions and has less restriction to the smoothness of the functions. The convergence rates and asymptotic distributions of the robust wavelet-based estimator are established when the design variables are stationary short-range dependent (SRD) and the errors are long-range dependent (LRD). Particularly, a rate of convergence [Formula: see text] in terms of estimation consistency can be achievable when the true components satisfy certain smoothness for a LRD process. Furthermore, an asymptotic property of the proposed estimator is given to indicate the confidence level of our proposed method for varying coefficient models with LRD.

Energy consumption and GDP: a panel data analysis with multi-level cross-sectional dependence

A fractionally integrated panel data model with a multi-level cross-sectional dependence is proposed. Such dependence is driven by a factor structure that captures comovements between blocks of variables through top-level factors, and within these blocks by non-pervasive factors. The model can include stationary and non-stationary variables, which makes it flexible enough to analyze relevant dynamics that are frequently found in macroeconomic and financial panels. The estimation methodology is based on fractionally differenced block-by-block cross-sectional averages. Monte Carlo simulations suggest that the procedure performs well in typical samples sizes. This methodology is applied to study the long-run relationship between energy consumption and economic growth. The main results suggest that estimates in some empirical studies may have some positive biases caused by neglecting the presence non-pervasive cross-sectional dependence and long-range dependence processes.

A note on stationary bootstrap variance estimator under long-range dependence

The stationary bootstrap method is popularly used to compute the standard errors or confidence regions of estimators, generated from time processes exhibiting weakly dependent stationarity. Most previous stationary bootstrap methods have focused on studying large-sample properties of stationary bootstrap inference about a sample mean under short-range dependence. For long-range dependence, recent studies have investigated the properties of block bootstrap methods using overlapping and non-overlapping blocking techniques with fixed block lengths. However, the characteristics of a stationary bootstrap with random block lengths are less well-known under long-range dependence. We investigate the asymptotic property of a stationary bootstrap variance estimator for a sample mean under long-range dependence. Our theoretical and simulation results indicate that the stationary bootstrap method does not have n−consistency for stationary and long-range dependent time processes.

A Comparison of Hurst Exponent Estimators in Long-range Dependent Curve Time Series

Abstract The Hurst exponent is the simplest numerical summary of self-similar long-range dependent stochastic processes. We consider the estimation of Hurst exponent in long-range dependent curve time series. Our estimation method begins by constructing an estimate of the long-run covariance function, which we use, via dynamic functional principal component analysis, in estimating the orthonormal functions spanning the dominant sub-space of functional time series. Within the context of functional autoregressive fractionally integrated moving average (ARFIMA) models, we compare finite-sample bias, variance and mean square error among some time- and frequency-domain Hurst exponent estimators and make our recommendations.

Stochastic differential equations with a fractionally filtered delay: A semimartingale model for long-range dependent processes

In this paper, we introduce a model, the stochastic fractional delay differential equation (SFDDE), which is based on the linear stochastic delay differential equation and produces stationary processes with hyperbolically decaying autocovariance functions. The model departs from the usual way of incorporating this type of long-range dependence into a short-memory model as it is obtained by applying a fractional filter to the drift term rather than to the noise term. The advantages of this approach are that the corresponding long-range dependent solutions are semimartingales and the local behavior of the sample paths is unaffected by the degree of long memory. We prove existence and uniqueness of solutions to the SFDDEs and study their spectral densities and autocovariance functions. Moreover, we define a subclass of SFDDEs which we study in detail and relate to the well-known fractionally integrated CARMA processes. Finally, we consider the task of simulating from the defining SFDDEs.

Frequency domain bootstrap for ratio statistics under long-range dependence

A frequency domain bootstrap (FDB) is a common technique to apply Efron’s independent and identically distributed resampling technique (Efron, 1979) to periodogram ordinates – especially normalized periodogram ordinates – by using spectral density estimates. The FDB method is applicable to several classes of statistics, such as estimators of the normalized spectral mean, the autocorrelation (but not autocovariance), the normalized spectral density function, and Whittle parameters. While this FDB method has been extensively studied with respect to short-range dependent time processes, there is a dearth of research on its use with long-range dependent time processes. Therefore, we propose an FDB methodology for ratio statistics under long-range dependence, using semi- and nonparametric spectral density estimates as a normalizing factor. It is shown that the FDB approximation allows for valid distribution estimation for a broad class of stationary, long-range (or short-range) dependent linear processes, without any stringent assumptions on the distribution of the underlying process. The results of a large simulation study show that the FDB approximation using a semi- or nonparametric spectral density estimator is often robust for various values of a long-memory parameter reflecting magnitude of dependence. We apply the proposed procedure to two data examples.

Data-driven semi-parametric detection of multiple changes in long-range dependent processes

This paper is devoted to the offline multiple changes detection for long-range dependence processes. The observations are supposed to satisfy a semi-parametric long-range dependence assumption with distinct memory parameters on each stage. A penalized local Whittle contrast is considered for estimating all the parameters, notably the number of changes. The consistency as well as convergence rates are obtained. Monte-Carlo experiments exhibit the accuracy of the estimators. They also show that the estimation of the number of breaks is improved by using a data-driven slope heuristic procedure of choice of the penalization parameter.

The function-indexed sequential empirical process under long-range dependence

Let $(\boldsymbol{X}_{j})_{j\geq1}$ be a multivariate long-range dependent Gaussian process. We study the asymptotic behavior of the corresponding sequential empirical process indexed by a class of functions. If some entropy condition is satisfied we have weak convergence to a linear combination of Hermite processes.

Estimation of slowly time-varying trend function in long memory regression models

ABSTRACTWe study the asymptotic properties of the least-squares estimator for the trend function of a particular class of locally stationary models, which are defined by considering a smooth variation of the trend function. Additionally, errors are assumed to be realizations from a long-range dependent stationary Gaussian process. Our findings are then illustrated through Monte Carlo simulations.

Convergence rates in the law of large numbers for long-range dependent linear processes

Baum and Katz (Trans. Am. Math. Soc. 120:108-123, 1965) obtained convergence rates in the Marcinkiewicz-Zygmund law of large numbers. Their result has already been extended to the short-range dependent linear processes by many authors. In this paper, we extend the result of Baum and Katz to the long-range dependent linear processes. As a corollary, we obtain convergence rates in the Marcinkiewicz-Zygmund law of large numbers for short-range dependent linear processes.

Large-sample approximations for variance-covariance matrices of high-dimensional time series

Distributional approximations of (bi--) linear functions of sample variance-covariance matrices play a critical role to analyze vector time series, as they are needed for various purposes, especially to draw inference on the dependence structure in terms of second moments and to analyze projections onto lower dimensional spaces as those generated by principal components. This particularly applies to the high-dimensional case, where the dimension $d$ is allowed to grow with the sample size $n$ and may even be larger than $n$. We establish large-sample approximations for such bilinear forms related to the sample variance-covariance matrix of a high-dimensional vector time series in terms of strong approximations by Brownian motions. The results cover weakly dependent as well as many long-range dependent linear processes and are valid for uniformly $ \ell_1 $-bounded projection vectors, which arise, either naturally or by construction, in many statistical problems extensively studied for high-dimensional series. Among those problems are sparse financial portfolio selection, sparse principal components, the LASSO, shrinkage estimation and change-point analysis for high--dimensional time series, which matter for the analysis of big data and are discussed in greater detail.

An [formula omitted]-estimator for the long-memory parameter

This paper proposes an M-estimator for the fractional parameter of stationary long-range dependent processes as an alternative to the classical GPH (Geweke and Porter-Hudak, 1983) method. Under very general assumptions on the long-range dependent process the consistency and the asymptotic normal distribution are established for the proposed method. One of the main results is that the convergence rate of the M-estimator is Nβ/2, for some positive β, which is the same rate as the standard GPH estimator. The asymptotic properties of the M-estimation method is investigated through Monte-Carlo simulations under the scenarios of ARFIMA models using contaminated with additive outliers and outlier-free data. The GPH approach is also considered in the study for comparison purposes, since this method is widely used in the literature of long-memory time series. The empirical investigation shows that M and GPH-estimator methods display standardized densities fairly close to the standard Gaussian density in the context of non-contaminated data. On the other hand, in the presence of additive outliers, the M-estimator remains unaffected with the presence of additive outliers while the GPH is totally corrupted, which was an expected performance of this estimator. Therefore, the M-estimator here proposed becomes an alternative method to estimate the long-memory parameter when dealing with long-memory time series with and without outliers.

Bootstrap methods for long-memory processes: a review

This manuscript summarized advances in bootstrap methods for long-range dependent time series data. The stationary linear long-memory process is briefly described, which is a target process for bootstrap methodologies on time-domain and frequency-domain in this review. We illustrate time-domain bootstrap under long-range dependence, moving or non-overlapping block bootstraps, and the autoregressive-sieve bootstrap. In particular, block bootstrap methodologies need an adjustment factor for the distribution estimation of the sample mean in contrast to applications to weak dependent time processes. However, the autoregressive-sieve bootstrap does not need any other modification for application to long-memory. The frequency domain bootstrap for Whittle estimation is provided using parametric spectral density estimates because there is no current nonparametric spectral density estimation method using a kernel function for the linear long-range dependent time process.

Measuring stationarity in long-memory processes

In this paper we consider the problem of measuring stationarity in locally stationary longmemory processes. We introduce an L2-distance between the spectral density of the locally stationary process and its best approximation under the assumption of stationarity. The distance is estimated by a numerical approximation of the integrated spectral periodogram and asymptotic normality of the resulting estimate is established. The results can be used to construct a simple test for the hypothesis of stationarity in locally stationary long-range dependent processes. We also propose a bootstrap procedure to improve the approximation of the nominal level and prove its consistency. Throughout the paper, we will work with Riemann sums of a squared periodogram instead of integrals (as it is usually done in the literature) and as a byproduct of independent interest it is demonstrated that the two approaches behave differently in the limit. AMS subject classification: 62M10, 62M15, 62G10