Long-range Dependent Processes Research Articles

Change-point tests under local alternatives for long-range dependent processes

We consider the change-point problem for the marginal distribution of subordinated Gaussian processes that exhibit long-range dependence. The asymptotic distributions of Kolmogorov-Smirnov- and Cramér-von Mises type statistics are investigated under local alternatives. By doing so we are able to compute the asymptotic relative efficiency of the mentioned tests and the CUSUM test. In the special case of a mean-shift in Gaussian data it is always $1$. Moreover, our theory covers the scenario where the Hermite rank of the underlying process changes. In a small simulation study, we show that the theoretical findings carry over to the finite sample performance of the tests.

Measuring stationarity in long-memory processes

In this paper we consider the problem of measuring stationarity in locally stationary long-memory processes. We introduce an $L_2$-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 by-product of independent interest it is demonstrated that the two approaches behave differently in the limit.

Statistics of Regional Surface Temperatures after 1900: Long-Range versus Short-Range Dependence and Significance of Warming Trends

AbstractThis paper studies regional climate variability for the time period 1900–2013 using parsimonious stochastic models. Instrumental data records on 5° × 5°, 2° × 2°, and equal-area grids are examined. A long-range dependent (LRD) stochastic process is used as a simplified description of the multitude of response times in the climate system. Fitting a linear trend to the global mean surface temperature (GMST) implies a warming of 0.08 decade−1, which is highly significant under an LRD null hypothesis (p < 10−4). The regional trends are distributed around the global mean trend, while the fluctuation levels increases when going from global to regional scale. The temperature fluctuations of the tropical oceans are observed to be strongly influenced by El Niño–Southern Oscillation (ENSO) and, therefore, more consistent with autoregressive processes of order 1 [AR(1)]. A likelihood-ratio test is used to systematically determine the best null model [AR(1) or LRD]. About 80% of the regional warming trends are found to be significant (with a 5% significance level).

Long memory and ARFIMA modelling: The case of CPI inflation rate in Ghana

In macroeconomic theory and economic policy, changes in the general price level or the rate of inflation plays an essential role. Hence, for example, one of the motives behind the adoption of Inflation Targeting policy (IT) by Ghana and the treaty espoused by the European Monetary Union, known as the Maastricht Treaty, was the convergence of inflation rates. On the other hand there is a controversy about which is the order of integration in the inflation rates, some authors arguing that this variable is stationary I(0) Whittle others saying it is nonstationary I(1). In this study we examine the CPI inflation rates of Ghana from a different perspective allowing for fractional degrees of differentiation. Thus, the methodology is based on long memory or long-range dependence processes, using fractional integration and employing techniques based on Whittle parametric and semiparametric methods and autoregressive fractionally integrated moving average (ARFIMA) models. Standard I(0)/I(1) methods were also employed. Our findings indicate that long memory exists in the CPI inflation rate of Ghana. After processing fractional differencing and determining the short memory components, the following two models, ARFIMA(3,0.427,1) and ARFIMA(2,0.499,1) were respectively specified to describe the pre and post introduction of IT policy in May 2007. Consequently, the CPI inflation rate of Ghana is fractionally integrated and mean reverting. Long memory in financial time series has important implications for the critical explanation of financial time series behaviour, as it could provide an opportunity to earn speculative profits in financial markets and cast disbelief on the correctness of the EMH. For instance, when price changes exhibit long memory or long-range dependence, asset pricing models based on the Efficient Market Hypothesis (EMH) may overestimate or underestimate investment risk. Furthermore, the presence of long memory in inflation rates can provide vital information about the likely impact of shocks (e.g. demand/supply) on the economy with respect to time. The results obtained in this study would be very useful in setting up monetary policies or consolidating previous policies such as IT in order to enhance economic growth. Moreover, estimation of long memory in inflation rates can serve as an evaluation tool to assess the performance of monetary policy under different dispensations. Lastly, the presence of long memory can assist in identifying inflationary pressures in the economy.

Intrinsic vs. spurious long-range memory in high-frequency records of environmental radioactivity

The presence or absence of long-range correlations in the environmental radioactivity fluctuations has recently attracted considerable interest. Among a multiplicity of practically relevant applications, identifying and disentangling the environmental factors controlling the variable concentrations of the radioactive noble gas radon is important for estimating its effect on human health and the efficiency of possible measures for reducing the corresponding exposition. In this work, we present a critical re-assessment of a multiplicity of complementary methods that have been previously applied for evaluating the presence of long-range correlations and fractal scaling in environmental radon variations with a particular focus on the specific properties of the underlying time series. As an illustrative case study, we subsequently re-analyze two high-frequency records of indoor radon concentrations from Coimbra, Portugal, each of which spans several weeks of continuous measurements at a high temporal resolution of five minutes.Our results reveal that at the study site, radon concentrations exhibit complex multi-scale dynamics with qualitatively different properties at different time-scales: (i) essentially white noise in the high-frequency part (up to time-scales of about one hour), (ii) spurious indications of a non-stationary, apparently long-range correlated process (at time scales between some hours and one day) arising from marked periodic components, and (iii) low-frequency variability indicating a true long-range dependent process. In the presence of such multi-scale variability, common estimators of long-range memory in time series are prone to fail if applied to the raw data without previous separation of time-scales with qualitatively different dynamics.

Weak convergence of the weighted sequential empirical process of some long-range dependent data

Let (Xk)k≥1 be a Gaussian long-range dependent process with EX1=0, EX12=1 and covariance function r(k)=k−DL(k). For any measurable function G let (Yk)k≥1=(G(Xk))k≥1. We study the asymptotic behaviour of the associated sequential empirical process (RN(x,t)) with respect to a weighted sup-norm ‖⋅‖w. We show that, after an appropriate normalization, (RN(x,t)) converges weakly in the space of cádlág functions with finite weighted norm to a Hermite process.

WHY FARIMA MODELS ARE BRITTLE

The FARIMA models, which have long-range-dependence (LRD), are widely used in many areas. Through the derivation of a precise characterization of the spectrum and variance time function, we show that this family is very atypical among LRD processes, being extremely close to the fractional Gaussian noise in a precise sense which results in ultra-fast convergence to fGn under rescaling. Furthermore, we show that this closeness property is not robust to additive noise. We argue that the use of FARIMA, and more generally fractionally differenced time series, should be reassessed in some contexts, in particular when convergence rate under rescaling is important and noise is expected.

Drift Parameter Estimation for a Reflected Fractional Brownian Motion Based on its Local Time

We consider a drift parameter estimation problem when the state process is a reflected fractional Brownian motion (RFBM) with a nonzero drift parameter and the observation is the associated local time process. The RFBM process arises as the key approximating process for queueing systems with long-range dependent and self-similar input processes, where the drift parameter carries the physical meaning of the surplus service rate and plays a central role in the heavy-traffic approximation theory for queueing systems. We study a statistical estimator based on the cumulative local time process and establish its strong consistency and asymptotic normality.

Drift Parameter Estimation for a Reflected Fractional Brownian Motion Based on its Local Time

We consider a drift parameter estimation problem when the state process is a reflected fractional Brownian motion (RFBM) with a nonzero drift parameter and the observation is the associated local time process. The RFBM process arises as the key approximating process for queueing systems with long-range dependent and self-similar input processes, where the drift parameter carries the physical meaning of the surplus service rate and plays a central role in the heavy-traffic approximation theory for queueing systems. We study a statistical estimator based on the cumulative local time process and establish its strong consistency and asymptotic normality.

Block sampling under strong dependence

The paper considers the block sampling method for long-range dependent processes. Our theory generalizes earlier ones by Hall et al. (1998) [11] on functionals of Gaussian processes and Nordman and Lahiri (2005) [16] on linear processes. In particular, we allow nonlinear transforms of linear processes. Under suitable conditions on physical dependence measures, we prove the validity of the block sampling method. Its finite-sample performance is illustrated by a simulation study.

Goodness-of-fit tests in long-range dependent processes under fixed alternatives

In a recent paper Fay and Philippe [ESAIM: PS 6 (2002) 239–258] proposed a goodness-of-fit test for long-range dependent processes which uses the logarithmic contrast as information measure. These authors established asymptotic normality under the null hypothesis and local alternatives. In the present note we extend these results and show that the corresponding test statistic is also normally distributed under fixed alternatives.

Power Spectrum of Generalized Fractional Gaussian Noise

Recently, we introduced a type of autocorrelation function (ACF) to describe a long-range dependent (LRD) process indexed with two parameters, which takes standard fractional Gaussian noise (fGn for short) as a special case. For simplicity, we call it the generalized fGn (GfGn). This short paper gives the power spectrum density function (PSD) of GfGn.

Fractional Black-Scholes Model and Technical Analysis of Stock Price

In the stock market, some popular technical analysis indicators (e.g., Bollinger bands, RSI, ROC, etc.) are widely used to forecast the direction of prices. The validity is shown by observed relative frequency of certain statistics, using the daily (hourly, weekly, etc.) stock prices as samples. However, those samples are not independent. In earlier research, the stationary property and the law of large numbers related to those observations under Black-Scholes stock price model and stochastic volatility model have been discussed. Since the fitness of both Black-Scholes model and short-range dependent process has been questioned, we extend the above results to fractional Black-Scholes model with Hurst parameterH>1/2, under which the stock returns follow a kind of long-range dependent process. We also obtain the rate of convergence.

Effects of trends and seasonalities on robustness of the Hurst parameter estimators

Long-range dependence (LRD) is discovered in time series arising from different fields, especially in network traffic and econometrics. Detecting the presence and the intensity of LRD plays a crucial role in time-series analysis and fractional system identification. The existence of LRD is usually indicated by the Hurst parameters. Up to now, many Hurst parameter estimators have been proposed in order to identify the LRD property involved in a time series. Since different estimators have different accuracy and robustness performances, in this study, 13 most popular Hurst parameter estimators are summarised and their estimation performances are investigated. LRD processes with known Hurst parameters are generated as the control data set for the robustness evaluation. In addition, three types of LRD processes are also obtained as the test signals by adding noises in terms of means, trends and seasonalities to the control data set. All 13 Hurst parameter estimators are applied to these LRD processes to estimate the existing Hurst parameters. The estimation results are documented and quantified by the standard errors. Conclusions of the accuracy and robustness performances of the estimators are drawn by comparing the estimation results.

A frequency domain bootstrap for Whittle estimation under long-range dependence

Whittle estimation is a common technique for fitting parametric spectral density functions to time series, in an effort to model the underlying covariance structure. However, Whittle estimators from long-range dependent processes can exhibit slow convergence to their Gaussian limit law so that calibrating confidence intervals with normal approximations may perform poorly. As a remedy, we study a frequency domain bootstrap (FDB) for approximating the distribution of Whittle estimators. The method provides valid distribution estimation for a broad class of stationary, long-range (or short-range) dependent linear processes, without stringent assumptions on the distribution of the underlying process. A large simulation study shows that the FDB approximations often improve normal approximations for setting confidence intervals for Whittle parameters in spectral models with strong dependence.

The sample autocorrelation function and the detection of long-memory processes

The detection of long-range dependence in time series analysis is an important task to which this paper contributes by showing that whilst the theoretical definition of a long-memory (or long-range dependent) process is based on the autocorrelation function, it is not possible for long memory to be identified using the sum of the sample autocorrelations, as usually defined. The reason for this is that the sample sum is a predetermined constant for any stationary time series; a result that is independent of the sample size. Diagnostic or estimation procedures, such as those in the frequency domain, that embed this sum are equally open to this criticism. We develop this result in the context of long memory, extending it to the implications for the spectral density function and the variance of partial sums of a stationary stochastic process. The results are further extended to higher order sample autocorrelations and the bispectral density. The corresponding result is that the sum of the third order sample (auto) bicorrelations at lags h,k≥1, is also a predetermined constant, different from that in the second order case, for any stationary time series of arbitrary length.

How are rescaled range analyses affected by different memory and distributional properties? A Monte Carlo study

In this paper, we present the results of Monte Carlo simulations for two popular techniques of long-range correlation detection — classical and modified rescaled range analyses. A focus is put on an effect of different distributional properties on an ability of the methods to efficiently distinguish between short-term memory and long-term memory. To do so, we analyze the behavior of the estimators for independent, short-range dependent, and long-range dependent processes with innovations from eight different distributions. We find that apart from a combination of very high levels of kurtosis and skewness, both estimators are quite robust to distributional properties. Importantly, we show that R/S is biased upwards (yet not strongly) for short-range dependent processes, while M-R/S is strongly biased downwards for long-range dependent processes regardless of the distribution of innovations.

A regularised estimator for long-range dependent processes

There is significant interest in long-range dependent processes since they occur in a wide range of phenomena across different areas of study. Based on the available models capable of describing long-range dependence, various parameter estimation methods have been developed. In this paper we revisit the maximum likelihood estimator and its computationally efficient approximations: Whittle’s Estimator and the Circulant Embedding Estimator. In particular, this paper proves the asymptotic properties of the Circulant Embedding estimator and establishes the asymptotic equivalence between the three estimators. Furthermore, it is shown that the three methods are ill-conditioned and thus highly sensitive to the presence of measurement errors. Finally, we introduce a regularisation method that improves the performance of the maximum likelihood methods when the datasets have been largely contaminated with errors.

INFERENCES OF -STABLE DISTRIBUTION OF THE UNDERLYING NOISE COMPONENTS IN GEODETIC DATA

We investigate the use of distribution functions to characterise the geophysical signals and noise components embedded in the geodetic Very Long Baseline Interferometry (VLBI) data sets across some of the International VLBI Service (IVS) stations. The rationale of using α-stable distributions as a tool to model the noise components in geodetic observables is due to the existence of impulsive signals/noise bursts (which often take the form of excursions with intermittent occurrences) in the data sets suggesting deviations from Gaussian distribution. A deviation from Gaussian distribution type would therefore suggest that statistical techniques such as least squares analysis, often used for analyzing the geodetic data (which are often based on Gaussian assumptions) could not be robust. In this paper, the properties of a long-range α-stable distribution with long tails and infinite moments in geodetic data are investigated by way of statistically testing their distribution using a family of stable distributions. The choice of stable distributions is based on the ease with which the statistical properties of the non-Gaussian processes are defined. Results indicate that the independent geophysical noise components reconstructed from geodetic VLBI baseline data exhibit distributions that have asymptotic power-law decay (albeit variable power indices) whose underlying process can be modelled as a long-range dependent process with an α-stable distribution (i.e., the stable varieties have small characteristic exponents).

Wavelet-based analysis of non-Gaussian long-range dependent processes and estimation of the hurst parameter

In this contribution, the statistical performance of the wavelet-based estimation procedure for the Hurst parameter is studied for non-Gaussian long-range dependent processes obtained from point transformations of Gaussian processes. The statistical properties of the wavelet coefficients and the estimation performance are compared both for processes having the same covariance but different marginal distributions and for processes having the same covariance and same marginal distributions but obtained from different point transformations, analyzed using mother wavelets with different number of vanishing moments. It is shown that the reduction of the dependence range from long to short by increasing the number of vanishing moments, observed for Gaussian processes, and at the origin of the popularity of the wavelet-based estimator, does not hold in general for non-Gaussian processes. Crucially, it is also observed that the Hermite rank of the point transformation impacts significantly the statistical properties of the wavelet coefficients and the estimation performance and also that processes having identical marginal distributions and covariance function can yet yield significantly different estimation performance. These results are interpreted in the light of central and noncentral limit theorems that are fundamental when dealing with long-range dependent processes. Moreover, it will be shown that, on condition that estimation is performed using a range of scales restricted to the coarsest practically available, an approximate, yet analytical and simple to use in practice, formula can be proposed for the evaluation of the variance of the wavelet-based estimator of the Hurst parameter.