Estimating the memory parameter for potentially non-linear and non-Gaussian time series with wavelets

The asymptotic theory for the memory-parameter estimator constructed from the log-regression with wavelets is incomplete for 1/$f$ processes that are not necessarily Gaussian or linear. Having a complete version of this theory is necessary because of the importance of non-Gaussian and non-linear long-memory models in describing financial time series. To bridge this gap, we prove that, under some mild assumptions, a newly designed memory estimator, named LRMW in this paper, is asymptotically consistent. The performances of LRMW in three simulated long-memory processes indicate the efficiency of this new estimator.

The Optimal Bandwidth Parameter Selection in GPH Estimation

In this paper, the optimal bandwidth parameter is investigated in the GPH algorithm. Firstly, combining with the stylized facts of financial time series, we generate long memory sequences by using the ARFIMA (1, d, 1) process. Secondly, we use the Monte Carlo method to study the impact of the GPH algorithm on existence test, persistence or antipersistence judgment of long memory, and the estimation accuracy of the long memory parameter. The results show that the accuracy of above three factors in the long memory test reached a relatively high level within the bandwidth parameter interval of 0.5 < a < 0.7. For different lengths of time series, bandwidth parameter a = 0.6 can be used as the optimal choice of the GPH estimation. Furthermore, we give the calculation accuracy of the GPH algorithm on existence, persistence or antipersistence of long memory, and long memory parameter d when a = 0.6.

A harmonically weighted filter for cyclical long memory processes

The estimation of the long memory parameter d is a widely discussed issue in the literature. The harmonically weighted (HW) process was recently introduced for long memory time series with an unbounded spectral density at the origin. In contrast to the most famous fractionally integrated process, the HW approach does not require the estimation of the d parameter, but it may be just as able to capture long memory as the fractionally integrated model, if the sample size is not too large. Our contribution is a generalization of the HW model, denominated the Generalized harmonically weighted (GHW) process, which allows for an unbounded spectral density at $$k \ge 1$$ k ≥ 1 frequencies away from the origin. The convergence in probability of the Whittle estimator is provided for the GHW process, along with a discussion on simulation methods. Fit and forecast performances are evaluated via an empirical application on paleoclimatic data. Our main conclusion is that the above generalization is able to model long memory, as well as its classical competitor, the fractionally differenced Gegenbauer process, does. In addition, the GHW process does not require the estimation of the memory parameter, simplifying the issue of how to disentangle long memory from a (moderately persistent) short memory component. This leads to a clear advantage of our formulation over the fractional long memory approach.

Fast computation and practical use of amplitudes at non-Fourier frequencies

In this paper, it is shown that the performance of various frequency-domain estimators of the memory parameter can be boosted by the inclusion of non-Fourier frequencies in addition to the regular Fourier frequencies. A fast two-stage algorithm for the efficient computation of the amplitudes at these additional frequencies is presented. In the first stage, the naïve sine and cosine transforms are computed with a modified version of the Fast Fourier Transform. In the second stage, these transforms are amended by taking the violation of the standard orthogonality conditions into account. A considerable number of auxiliary quantities, which are required in the second stage, do not depend on the data and therefore only need to be computed once. The superior performance (in terms of root-mean-square error) of the estimators based also on non-Fourier frequencies is demonstrated by extensive simulations. Finally, the empirical results obtained by applying these estimators to financial high-frequency data show that significant long-range dependence is present only in the absolute intraday returns but not in the signed intraday returns.

Local Whittle estimation of the long-memory parameter

In this article, we describe and implement the local Whittle and exact local Whittle estimators of the order of fractional integration of a time series.

Frequency-Domain Evidence for Climate Change

The goal of this paper is to search for conclusive evidence against the stationarity of the global air surface temperature, which is one of the most important indicators of climate change. For this purpose, possible long-range dependencies are investigated in the frequency-domain. Since conventional tests of hypotheses about the memory parameter, which measures the degree of long-range dependence, are typically based on asymptotic arguments and are therefore of limited practical value in case of small or medium sample sizes, we employ a new small-sample test as well as a related estimator for the memory parameter. To safeguard against false positive findings, simulation studies are carried out to examine the suitability of the employed methods and hemispheric datasets are used to check the robustness of the empirical findings against low-frequency natural variability caused by oceanic cycles. Overall, our frequency-domain analysis provides strong evidence of non-stationarity, which is consistent with previous results obtained in the time domain with models allowing for stochastic or deterministic trends.