Long Memory Time Series Research Articles

Burn-in selection in simulating stationary time series

Many time series models are defined in a recursive manner, which prohibits exact simulations. In practice, one appeals to simulating a long time series and discarding a large number of initial simulated observations, known as the burn-in. For autoregressive models where the dependence decays exponentially fast, the choice of the burn-in is not critical. However, for long-memory time series where the dependence from the remote past is strong, it is not clear how to select the burn-in number. By combining several samplers with randomized burn-in numbers, a method for exactly simulating the expectation of a statistic computed from a time series is developed. Moreover, with some suitably chosen statistics, the exact simulation method can be applied to quantify the effect of burn-in numbers on the simulated sample. Simulation studies are conducted to provide some practical guidances for burn-in selections.

Predicting the Long-Term Dependencies in Time Series Using Recurrent Artificial Neural Networks

Long-term dependence is an essential feature for the predictability of time series. Estimating the parameter that describes long memory is essential to describing the behavior of time series models. However, most long memory estimation methods assume that this parameter has a constant value throughout the time series, and do not consider that the parameter may change over time. In this work, we propose an automated methodology that combines the estimation methodologies of the fractional differentiation parameter (and/or Hurst parameter) with its application to Recurrent Neural Networks (RNNs) in order for said networks to learn and predict long memory dependencies from information obtained in nonlinear time series. The proposal combines three methods that allow for better approximation in the prediction of the values of the parameters for each one of the windows obtained, using Recurrent Neural Networks as an adaptive method to learn and predict the dependencies of long memory in Time Series. For the RNNs, we have evaluated four different architectures: the Simple RNN, LSTM, the BiLSTM, and the GRU. These models are built from blocks with gates controlling the cell state and memory. We have evaluated the proposed approach using both synthetic and real-world data sets. We have simulated ARFIMA models for the synthetic data to generate several time series by varying the fractional differentiation parameter. We have evaluated the proposed approach using synthetic and real datasets using Whittle’s estimates of the Hurst parameter classically obtained in each window. We have simulated ARFIMA models in such a way that the synthetic data generate several time series by varying the fractional differentiation parameter. The real-world IPSA stock option index and Tree Ringtime series datasets were evaluated. All of the results show that the proposed approach can predict the Hurst exponent with good performance by selecting the optimal window size and overlap change.

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.

Confidence Regions for Parameters in Stationary Time Series Models With Gaussian Noise

This article develops two new empirical likelihood methods for long-memory time series models based on adjusted empirical likelihood and mean empirical likelihood. By application of Whittle likelihood, one obtains a score function that can be viewed as the estimating equation of the parameters of the long-memory time series model. An empirical likelihood ratio is obtained which is shown to be asymptotically chi-square distributed. It can be used to construct confidence regions. By adding pseudo samples, we simultaneously eliminate the non-definition of the original empirical likelihood and enhance the coverage probability. Finite sample properties of the empirical likelihood confidence regions are explored through Monte Carlo simulation, and some real data applications are carried out.

Monitoring Mean and Variance Change-Points in Long-Memory Time Series

Monitoring Mean and Variance Change-Points in Long-Memory Time Series

Integration and Disintegration of EMU Government Bond Markets

It is commonly found that the markets for long-term government bonds of Economic and Monetary Union (EMU) countries were integrated prior to the EMU debt crisis. Contrasting this, we show, based on the interrelation between market integration and fractional cointegration, that there were periods of integration and disintegration that coincide with bull and bear market periods in the stock market. An econometric argument about the spectral behavior of long-memory time series leads to the conclusion that there is a stronger differentiation between bonds with different default risks. This implied the possibility of macroeconomic and fiscal divergence between the EMU countries before the crisis periods.

LM Tests for Joint Breaks in the Dynamics and Level of a Long-Memory Time Series

We consider a single-step Lagrange multiplier (LM) test for joint breaks (at known or unknown dates) in the long memory parameter, the short-run dynamics, and the level of a fractionally integrated time-series process. The regression version of this test is easily implementable and allows to identify the specific sources of the break when the null hypothesis of parameter stability is rejected. However, its size and power properties are sensitive to the correct specification of short-run dynamics under the null. To address this problem, we propose a slight modification of the LM test (labeled LMW-type test) which also makes use of some information under the alternative (in the spirit of a Wald test). This test shares the same limiting distribution as the LM test under the null and local alternatives but achieves higher power by facilitating the correct specification of the short-run dynamics under the null and any alternative (either local or fixed). Monte Carlo simulations provide support for these theoretical results. An empirical application, concerning the origin of shifts in the long-memory properties of forward discount rates in five G7 countries, illustrates the usefulness of the proposed LMW-type test.

Inference for Long-memory Time Series Models Based on Modified Empirical Likelihood

Empirical likelihood method has been applied to short-memory time series models by Monti (1997) through the Whittle's estimation method. Yau (2012) extended this idea to long-memory time series models. Asymptotic distributions of the empirical likelihood ratio statistic for short and long-memory time series have been derived to construct confidence regions for the corresponding model parameters. However, it experiences the undercoverage issue which causes the coverage probabilities of parameters lower than the given nominal levels, especially for small sample sizes. In this paper, we propose a modified empirical likelihood which combines the advantages of the adjusted empirical likelihood and the transformed empirical likelihood to modify the one proposed by Yau (2012) for autoregressive fractionally integrated moving average (ARFIMA) model for the purpose of improving coverage probabilities.Asymptotic null distribution of the test statistic has been established as the standard chi-square distribution with the degree of freedom one. Simulations have been conducted to investigate the performance of the proposed method as well as the comparisons of other existing methods to illustrate that the proposed method can provide better coverage probabilities especially for small sample sizes.

Forecasting of Price of Rice in India Using Long-Memory Time-Series Model

Price forecasting of agricultural commodities plays an important role in efficient planning and formulation of executive decisions. In this study, an attempt has been made to apply autoregressive fractionally integrated moving average (ARFIMA) model to the daily all India maximum, minimum and modal wholesale price data of rice in order to capture the observed long-run persistency. The price series under consideration are stationary, but there is a significant presence of long memory in the price data. Accordingly, ARFIMA model is applied to obtain the forecasts and window-based evaluation of forecasting is carried out with the help of relative mean absolute percentage error, root mean square error and mean absolute error. To this end, a comparative study has also been made between the best fitted ARFIMA model and the best fitted ARIMA model and it is observed that ARFIMA model outperforms the usual ARIMA model.

Reviewing monetary policy inertia and its effects: The fractional integration approach for an emerging economy

This paper addressed the monetary policy inertia of a relevant emerging economy based on the fractional integration approach (Geweke and Porter-Hudak, 1983; Baillie et al. 1996), so that we avoided the potential spurious estimates for such an inertia that can be caused by adopting conventional unit root tests, which were usually applied to the related literature. We thus confirmed the hypothesis of a high monetary policy inertia (Clarida et al., 1999), even after controlling for long memory in time series. Furthermore, we identified a positive relationship between inflationary expectations and the inertial component. It means that, under an inflation targeting regime, central banks should adjust both the basic interest rate and its inertial path in order to accomplish the inflation target over time.

A modified Wilcoxon test for change points in long-range dependent time series

This paper considers testing for structural change in long-memory time series. We modify the Wilcoxon two-sample rank test by standardizing it with a kernel-based fixed bandwidth long-run variance estimator. The corresponding test statistic converges to a well-defined distribution under the null hypothesis. In a Monte Carlo simulation we confirm that the test provides good finite sample size and power results and compare it with an existing approach.

Monitoring memory parameter change-points in long-memory time series

In this paper, we propose two ratio-type statistics to sequentially detect the memory parameter change-points in the long-memory time series. The limiting distributions of monitoring statistics under the no-change-point null hypothesis as well as their consistency under the alternative hypothesis are proved. In particular, a sieve bootstrap approximation method is proposed to determine the critical values. Extensive simulations indicate that the new monitoring procedures perform well in finite samples. Finally, we illustrate our monitoring procedures by two sets of real data.

Simultaneous quantile inference for non-stationary long-memory time series

We consider the simultaneous or functional inference of time-varying quantile curves for a class of non-stationary long-memory time series. New uniform Bahadur representations and Gaussian approximation schemes are established for a broad class of non-stationary long-memory linear processes. Furthermore, an asymptotic distribution theory is developed for the maxima of a class of non-stationary long-memory Gaussian processes. Using the latter theoretical results, simultaneous confidence bands for the aforementioned quantile curves with asymptotically correct coverage probabilities are constructed.

Change-in-mean tests in long-memory time series: a review of recent developments

It is well known that standard tests for a mean shift are invalid in long-range dependent time series. Therefore, several long-memory robust extensions of standard testing principles for a change-in-mean have been proposed in the literature. These can be divided into two groups: those that utilize consistent estimates of the long-run variance and self-normalized test statistics. Here, we review this literature and complement it by deriving a new long-memory robust version of the sup-Wald test. Apart from giving a systematic review, we conduct an extensive Monte Carlo study to compare the relative performance of these methods. Special attention is paid to the interaction of the test results with the estimation of the long-memory parameter. Furthermore, we show that the power of self-normalized test statistics can be improved considerably by using an estimator that is robust to mean shifts.

M-periodogram for the analysis of long-range-dependent time series

ABSTRACTThis paper focuses on time series with long-memory and suggests using an alternative periodogram, called M-periodogram, which is obtained by relating the periodogram to a regression problem and then using an M-estimator for the coefficients of the regression model. The asymptotic properties of this novel M-periodogram are established and its empirical properties are investigated for finite samples under different scenarios. Furthermore, in addition to being an appealing alternative periodogram for long-memory time series, it is also resistant to additive outliers. We investigate the robustness performance of the estimator through simulation. As a practical application, the paper investigates the effect of atypical observations in air pollution data, namely daily Particulate Matter () observations. Besides the importance of modelling and forecasting this pollutant, the series presents, in general, interesting features such as seasonal poles, asymmetry, and also high levels of pollution which can be regarded as atypical observations in the context of this work.

Harmonically Weighted Processes: The Case of U.S. Inflation

We suggest a model for long memory in time series that amounts to harmonically weighting short memory processes, ∑ j x t − j / ( j 1). A nonstandard rate of convergence is required to establish a Gaussian functional central limit theorem. Further, we study the asymptotic least squares theory when harmonically weighted processes are regressed on each other. The regression estimators converge to Gaussian limits upon the conventional normalization with square root of the sample size, and standard testing procedures apply. Harmonically weighted processes do not allow - or require - to choose a memory parameter. Nevertheless, they may well be able to capture dynamics that have been modelled by fractional integration in the past, and the conceptual simplicity of the new model may turn out to be a worthwhile advantage in practice. The harmonic inverse transformation that removes this kind of long memory is also developed. We successfully apply the procedure to monthly U.S. inflation, and provide simulation evidence that fractional integration of order d is well captured by harmonic weighting over a relevant range of d in finite samples.

S&P500 volatility analysis using high-frequency multipower variation volatility proxies

The availability of ultra-high-frequency data has sparked enormous parametric and nonparametric volatility estimators in financial time series analysis. However, some high-frequency volatility estimators are suffering from biasness issues due to the abrupt jumps and microstructure effect that often observed in nowadays global financial markets. Hence, we motivate our studies with two long-memory time series models using various high-frequency multipower variation volatility proxies. The forecast evaluations are illustrated using the S&P500 data over the period from year 2008 to 2013. Our empirical studies found that higher-power variation volatility proxies provide better in-sample and out-of-sample performances as compared to the widely used realized volatility and fractionally integrated ARCH models. Finally, these empirical findings are used to estimate the one-day-ahead value-at-risk of S&P500.

A simple test on structural change in long-memory time series

We propose a simple test on structural change in long-range dependent time series. It is based on the idea that the test statistic of the standard CUSUM test retains its asymptotic distribution if it is applied to fractionally differenced data. We prove that our approach is asymptotically valid, if the memory is estimated consistently under the null hypothesis. Therefore, the well-known CUSUM test can be used on the differenced data without any further modification. In a simulation study, we compare our test with a CUSUM test on structural change that is specifically constructed for long-memory time series and show that our approach performs reasonably well.

On the validity of resampling methods under long memory

For long-memory time series, inference based on resampling is of crucial importance, since the asymptotic distribution can often be non-Gaussian and is difficult to determine statistically. However, due to the strong dependence, establishing the asymptotic validity of resampling methods is nontrivial. In this paper, we derive an efficient bound for the canonical correlation between two finite blocks of a long-memory time series. We show how this bound can be applied to establish the asymptotic consistency of subsampling procedures for general statistics under long memory. It allows the subsample size $b$ to be $o(n)$, where $n$ is the sample size, irrespective of the strength of the memory. We are then able to improve many results found in the literature. We also consider applications of subsampling procedures under long memory to the sample covariance, M-estimation and empirical processes.

Robust estimation of fractional seasonal processes: Modeling and forecasting daily average SO2 concentrations

This paper deals with the estimation of seasonal long-memory time series models in the presence of ‘outliers’. It is long known that the presence of outliers can lead to undesirable effects on the statistical estimation methods, for example, substantially impacting the sample autocorrelations. Thus, the aim of this work is to propose a semiparametric robust estimator for the fractional parameters in the seasonal autoregressive fractionally integrated moving average (SARFIMA) model, through the use of a robust periodogram at both very low and seasonal frequencies. The model and some theories related to the estimation method are discussed. It is shown by simulations that the robust methodology behaves like the classical one to estimate the long-memory parameters if there are no outliers (no contamination). On the other hand, in the contaminated scenario (presence of outliers), the standard methodology leads to misleading results while the proposed method is unaffected. The methodology is applied to model and forecast sulfur dioxide (SO2) pollutant concentrations which have seasonal long-memory features and occasional large peak pollutant concentrations.