Long Memory Processes Research Articles

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

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.

Average run length of the long-memory autoregressive fractionally integrated moving average process of the exponential weighted moving average control chart

AbstractMeasurement of control chart efficiency by comparison of average run length (ARL) is widely implemented in quality control. The aim of this study is to evaluate the ARL, which is a solution of the integral equation obtained from the Exponential Weighted Moving Average (EWMA) statistic with a long-memory Autoregressive Fractionally Integrated Moving Average (ARFIMA) process. The derivation of the analytical ARL of the EWMA control chart and proof of the existence and uniqueness of the analytical ARL by Fixed Point theory are shown. Moreover, the numerical ARL carried out by the Composite Midpoint Rule technique of the EWMA control chart is demonstrated. A comparison between the analytical and numerical ARL is also illustrated. The findings indicated that analytical ARL of the EWMA control chart is more quickly computational than the numerical ARL. Therefore, the analytical ARL is an alternative method for measuring the efficiency of the EWMA control chart with the long-memory ARFIMA process.

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.

Modelling the volatility of commodities prices using a stochastic volatility model with random level shifts

We use the approach of Qu and Perron (Econom J 16(3):309–339, 2013) for the modeling and inference of volatility of a set of commodity prices in the presence of random level shifts of unknown timing, magnitude and frequency. Our approach contributes to the study of commodities in several aspects. First, we test for the presence of a genuine long-memory process in the volatility of commodities. Second, we determine that the random level shifts are certainly the main source of variation in the commodity price volatility. Finally, we estimate the volatility and its components as latent variables, thereby making it possible to evaluate their level of correlation with macroeconomic variables in small open economies such as Latin-American countries where the dependence on commodity price volatility is high. We use six commodity series: agriculture, livestock, gold, oil, industrial metals and a general commodity index. All series cover the period from January 1983 until December 2013 in daily frequency. The results show that although the occurrence of a level shift is rare, (about once every 1.5 or 1.8 years), this component clearly contributes most to the variation in the volatility. Furthermore, isolating the level shift component from the overall volatility indicates a strong relationship of this component with a set of business cycle indicators of several Latin American countries.

IMPROVED GENERATOR OF LONG-MEMORY PROCESSES

Simulations with long-memory input processes are hindered both by the slowness of convergence displayed by the output data and by the high computational complexity of the on-line methods for generating the input pro- cess. We present an optimized algorithm for simulating efficiently the occupancy process of the M/G/∞ system, which can be used as a sequential pseudo- random number generator of a broad class of long-memory sample paths. Our previous approach is the decomposition of the service time distribution as a lin- ear combination of memoryless random variables, plus a residual term. Then, the original M/G/∞ system is replaced by a number of parallel, independent, virtual and easier to simulate M/G/∞ subsystems, the dynamics of which can be replicated sequentially or in parallel too. In this work we improve our previ- ous algorithm, taking into account the generation time of the random variables.

Applicability of the Whittle estimator to non-stationary and non-linear long-memory processes

The Whittle estimator is a classic statistic to measure the long-memory parameter of stationary stochastic processes. Recently, the theoretical framework of the estimator has been extended to include its application to non-linear and non-stationary processes. The asymptotic behaviour of the generalized estimator has been analysed in several works, but there seems to be limited empirical studies about the robustness of the estimator on real or synthetic time series. In this paper, we test the robustness of the general Whittle estimator applied to some classes of non-stationary and non-linear long-memory processes. We evaluate the bias and variance of the estimator of the long-memory parameter for different combinations of the correlation parameters and the sample sizes. The numerical results obtained indicate that the performance of the estimator is good, but the sample length necessary to obtain good estimations depends on the type of process and the degree of short and long-term correlation.

Time-Varying Exchange Rate Exposure: Evidence From Emerging Markets

Using weekly data obtained from three emerging market economies, this study looks into the time-varying characteristic of exchange rate exposure coefficients. In doing so, unlike in some previous studies, exchange rate exposure is viewed through the eyes of an investor in the relevant country. The analysis is carried out using country-level stock indexes and trade-weighted exchange rates. Time-varying exchange rate exposure coefficients are obtained by estimating a multivariate BEKK-GARCH-M model with explicit focus on the non-orthogonality between exchange rate changes and market returns. The findings of the study indicate that, although they are likely to vary over time, exchange rate exposure coefficients of two out of the three cases follow mean-reverting long-memory processes. The presence of mean-reverting exchange rate exposure coefficients has important implications for investment and hedging strategies. In addition, time-varying exchange rate exposure coefficients turn out to be more volatile than respective market betas.

Spectral Characteristics of Instrumental and Climate Model Surface Temperatures

Abstract The spatiotemporal temperature variability for several gridded instrumental and general circulation climate model data is characterized, contrasting power spectra of local and global temperatures, land and sea temperatures, and temperatures of different regions. There is generally a high degree of agreement between the spectral characteristics of instrumental and climate model data. All but the equatorial spectra exhibit a power-law shape and are hence more consistent with the spectra expected from long-memory processes than from short-memory processes. The power-law exponent β of the spectra is a measure of memory, or persistence, of the temperatures and is observed to be about twice as large for global temperature than for local temperatures. However, there are large variations, in particular between land and sea surface temperatures. This is shown by estimates of the spectra for different regions and global maps of β. It is also demonstrated that global spectra are related to local spectra via teleconnections between local temperatures.

An integrated heteroscedastic autoregressive model for forecasting realized volatilities

A new strategy for forecasting realized volatility (RV) is proposed for the heteroscedastic autoregressive (HAR) model of Corsi (2009). The strategy is constraining the sum of the HAR coefficients to one, resulting in an integrated model, called IHAR model. The IHAR model is motivated by stationarity of estimated HAR model, downward biases of estimated HAR coefficients, and over-rejection of ADF test for long-memory processes. Considerable out-of-sample forecast improvements of the IHAR model over the HAR model are demonstrated for RVs of 4 financial assets: the US S&P 500 index, the US NASDAQ index, the Japan yen/US dollar exchange rate, and the EU euro/US dollar exchange rate. Forecast improvement is also verified in a Monte Carlo experiment and in an empirical comparison for an extended data set. The forecast improvement is shown to be a consequence of the fact that the IHAR model takes better advantage of the long memory of RV and the conditional heteroscedasticity of RV than the HAR model.

Semiparametric Stationarity and Fractional Unit Roots Tests Based on Data-Driven Multidimensional Increment Ratio Statistics

AbstractIn this paper, we show that the central limit theorem (CLT) satisfied by the data-driven Multidimensional Increment Ratio (MIR) estimator of the memory parameter

Wavelet-Tsallis Entropy Detection and Location of Mean Level-Shifts in Long-Memory fGn Signals

Long-memory processes, in particular fractional Gaussian noise processes, have been applied as models for many phenomena occurring in nature. Non-stationarities, such as trends, mean level-shifts, etc., impact the accuracy of long-memory parameter estimators, giving rise to biases and misinterpretations of the phenomena. In this article, a novel methodology for the detection and location of mean level-shifts in stationary long-memory fractional Gaussian noise (fGn) signals is proposed. It is based on a joint application of the wavelet-Tsallis q-entropy as a preprocessing technique and a peak detection methodology. Extensive simulation experiments in synthesized fGn signals with mean level-shifts confirm that the proposed methodology not only detects, but also locates level-shifts with high accuracy. A comparative study against standard techniques of level-shift detection and location shows that the technique based on wavelet-Tsallis q-entropy outperforms the one based on trees and the Bai and Perron procedure, as well.

Monte Carlo Maximum Likelihood Estimation for Generalized Long-Memory Time Series Models

An exact maximum likelihood method is developed for the estimation of parameters in a non-Gaussian nonlinear density function that depends on a latent Gaussian dynamic process with long-memory properties. Our method relies on the method of importance sampling and on a linear Gaussian approximating model from which the latent process can be simulated. Given the presence of a latent long-memory process, we require a modification of the importance sampling technique. In particular, the long-memory process needs to be approximated by a finite dynamic linear process. Two possible approximations are discussed and are compared with each other. We show that an autoregression obtained from minimizing mean squared prediction errors leads to an effective and feasible method. In our empirical study, we analyze ten daily log-return series from the S&P 500 stock index by univariate and multivariate long-memory stochastic volatility models. We compare the in-sample and out-of-sample performance of a number of models within the class of long-memory stochastic volatility models.

Nonlinear analysis of pupillary dynamics.

Pupil size reflects autonomic response to different environmental and behavioral stimuli, and its dynamics have been linked to other autonomic correlates such as cardiac and respiratory rhythms. The aim of this study is to assess the nonlinear characteristics of pupil size of 25 normal subjects who participated in a psychophysiological experimental protocol with four experimental conditions, namely “baseline”, “anger”, “joy”, and “sadness”. Nonlinear measures, such as sample entropy, correlation dimension, and largest Lyapunov exponent, were computed on reconstructed signals of spontaneous fluctuations of pupil dilation. Nonparametric statistical tests were performed on surrogate data to verify that the nonlinear measures are an intrinsic characteristic of the signals. We then developed and applied a piecewise linear regression model to detrended fluctuation analysis (DFA). Two joinpoints and three scaling intervals were identified: slope α0, at slow time scales, represents a persistent nonstationary long-range correlation, whereas α1 and α2, at middle and fast time scales, respectively, represent long-range power-law correlations, similarly to DFA applied to heart rate variability signals. Of the computed complexity measures, α0 showed statistically significant differences among experimental conditions (p<0.001). Our results suggest that (a) pupil size at constant light condition is characterized by nonlinear dynamics, (b) three well-defined and distinct long-memory processes exist at different time scales, and (c) autonomic stimulation is partially reflected in nonlinear dynamics.

EWMA control charts for detecting changes in the mean of a long-memory process

In this paper EWMA control charts are introduced for detecting changes in the mean of a long-memory process. Besides the modified EWMA scheme the EWMA residual chart is also considered. The control design of the charts is calculated for an ARFIMA(p,d,q) process. In order to assess the introduced charts, the average run length is used as a performance criterion. Using an extensive simulation study, the control charts are compared with each other. The target process is assumed to be an ARFIMA(1,d,1) process.

Long-memories and mean breaks in realized volatilities

An extended sequential test of Bai and Perron (1998) to a long-memory process is applied to four sets of realized volatilities (RVs) of the US dollar–EU euro, the Japan yen–US dollar, the Korea won–US dollar exchange rates and the S&P 500 index to find significant structural breaks in the means. Even after the mean breaks are adjusted out, the RVs still have persistent memories, which will be shown to produce better out-of-sample forecasts of RVs if properly addressed than ignored. Contrary to the recent report of Choi et al. (2010) that ‘short-memory + break’ models have better forecast power than ‘long-memory only’ models in forecasting some foreign exchange rate RVs, models with ‘long-memory + mean breaks’ turn out to produce better out-of-sample forecasts than models with ‘short-memory + mean breaks’ and models with ‘long-memory only’.

The dynamics of the epileptic brain reveal long-memory processes.

The pattern of epileptic seizures is often considered unpredictable and the interval between events without correlation. A number of studies have examined the possibility that seizure activity respects a power-law relationship, both in terms of event magnitude and inter-event intervals. Such relationships are found in a variety of natural and man-made systems, such as earthquakes or Internet traffic, and describe the relationship between the magnitude of an event and the number of events. We postulated that human inter-seizure intervals would follow a power-law relationship, and furthermore that evidence for the existence of a long-memory process could be established in this relationship. We performed a post hoc analysis, studying eight patients who had long-term (up to 2 years) ambulatory intracranial EEG data recorded as part of the assessment of a novel seizure prediction device. We demonstrated that a power-law relationship could be established in these patients (β = − 1.5). In five out of the six subjects whose data were sufficiently stationary for analysis, we found evidence of long memory between epileptic events. This memory spans time scales from 30 min to 40 days. The estimated Hurst exponents range from 0.51 to 0.77 ± 0.01. This finding may provide evidence of phase-transitions underlying the dynamics of epilepsy.

On functional limits of short- and long-memory linear processes with GARCH(1,1) noises

This paper considers the short- and long-memory linear processes with GARCH (1,1) noises. The functional limit distributions of the partial sum and the sample autocovariances are derived when the tail index α is in (0,2), equal to 2, and in (2,∞), respectively. The partial sum weakly converges to a functional of α-stable process when α<2 and converges to a functional of Brownian motion when α≥2. When the process is of short-memory and α<4, the autocovariances converge to functionals of α/2-stable processes; and if α≥4, they converge to functionals of Brownian motions. In contrast, when the process is of long-memory, depending on α and β (the parameter that characterizes the long-memory), the autocovariances converge to either (i) functionals of α/2-stable processes; (ii) Rosenblatt processes (indexed by β, 1/2<β<3/4); or (iii) functionals of Brownian motions. The rates of convergence in these limits depend on both the tail index α and whether or not the linear process is short- or long-memory. Our weak convergence is established on the space of càdlàg functions on [0,1] with either (i) the J1 or the M1 topology (Skorokhod, 1956); or (ii) the weaker form S topology (Jakubowski, 1997). Some statistical applications are also discussed.

Time-Varying Exchange Rate Exposure: Evidence from Emerging Markets

This paper examines exchange rate exposure of country level stock returns in three emerging market economies: Korea, Taiwan and Thailand. The analysis is carried out at country level using stock indexes and trade-weighted exchange rates. Time-varying exchange rate exposure coefficients are obtained by estimating a Multivariate GARCH-M model with explicit focus on the non-orthogonality between exchange rate changes and market returns. Findings of the paper indicate that, although they are likely to vary over time, exchange rate exposure coefficients for Korea and Taiwan follow mean-reverting long-memory processes. However, the exposure coefficient for Thailand is found to be characterized by a non-stationary unit root process. The presence of mean-reverting exchange rate exposure coefficients has important implications for investment and hedging strategies.