Short Memory in Long Memory Time Series

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Asymptotic properties and inference procedures for long memory processes have been extensively studied in the last 30 years. However, when a long memory process involves short memory components, statistical inference methods are insufficient and need to be substantially enhanced. Specifically, if a fractionally integrated autoregressive moving-average (ARFIMA) process contains autoregressive moving-average (ARMA) components, currently applied statistical methods frequently produce biases that result in significant inaccuracies. Accordingly, there is a need for a more accurate investigation of short memory components pertinent in the ARFIMA process. This project considers several statistical problems in time series settings where the data has both long memory and short memory characteristics. The statistical problems considered include: (1) testing to determine if a long memory time series has short memory characteristics; (2) developing stochastic parameter regression models of long memory and short memory characteristics with a simpler autocorrelation structure; and (3) assessing biases in the sample autocorrelations and cross-correlations for long memory time series with short memory characteristics.Studying long memory time series with short memory components is very important, as they are frequently observed in real-world contexts, such as stock returns and volatilities, inflation rates, temperatures, and river levels. The project aims to develop accurate statistical models and inference methods to analyze such time series. The development of this research will: (1) advance the theory and methods of long memory processes; (2) help the public better understand global warming issues with the proposed models and methods; and (3) benefit practitioners to use the research outcomes in their disciplines. In addition, the investigator will contribute to the launch of Boise State's mathematical and statistical consulting center. The center will be a hub of applied mathematics and statistics fused with other sciences, serving Boise and the State of Idaho where no such facility is currently available.