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.

Time Trends and Persistence in the Snowpack Percentages by Watershed in Colorado

In this paper we investigate the time trend coefficients in snowpack percentages by watershed in Colorado, US, allowing for the possibility of long range dependence or long memory processes. Nine series corresponding to the following watersheds are examined: Arkansas, Colorado, Gunnison, North Platte, Rio Grande, South Platte, San Juan-Animas-Dolores-San Miguel, Yampa & White and Colorado Statewide, based on annual data over the last eighty years. The longest series start in 1937 and all end in 2019. The results indicate that most of the series display a significant decline over time, showing negative time trend coefficients, and thus supporting the hypothesis of climate change and global warming. Nevertheless, there is no evidence of a long memory pattern in the data.

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.