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.

Project Report

Autocorrelations are an important information in modeling time series data, producing more realistic models and better prediction powers when identified appropriately. Short memory autocorrelations decrease fast when time series data are apart. In contrast, long memory autocorrelations persist for a long time and are not absolutely summable, implying that distant time series data are still substantially correlated. Due to this infinite summability of autocorrelations, many classical limit theorems for time series data do not immediately hold. While these theoretical difficulties are inherent, the long memory time series models are popular in many disciplines such as economics, finance, climatology, hydrology, and engineering. This project studies short memory characteristics in long memory time series data. We develop a new test for identifying short memory components in long memory time series. This test particularly finds short memory autocorrelations in United States inflation rate data, whereas existing tests fail to find. Modeling these short memory characteristics leads to improved model accuracy and more precise prediction. We also study the generalized least squares (GLS) estimators in time series settings, developing a new fast GLS algorithm for periodically changing times series. We also derive an explicit expression for the GLS estimators for a linear trend in the regression model with autoregressive errors and obtain more precise convergence rates of these least squares estimators, helping justify why that ordinary least squares methods have been successful in many time series settings. The models and methods developed in this project are applied to several economic data. Our long memory stochastic parameter regression model particularly fits well to explain Australian Consumer Price Index rate series with United States three month Treasury bill series. Also, we find that gold and silver prices are found to be well related with time-varying stochastic models. Development of the models and methods in this project also leads to impact on climatology. We develop rigorous trend estimation techniques for United States monthly extreme (maximum and minimum) temperatures. As mean and extreme statistics are probabilistically independent, there is no reason to assume that extreme temperatures have changed similarly to mean temperatures. Spacial maps of estimated trends in extreme temperatures are produced. Our results show that maximum temperatures are not significantly changing but minimum temperatures show significant warming. Changepoints times are also estimated and could be used for other extreme temperature studies. We also studied trends in extreme United States precipitation reanalysis products (MERRA and NARR) and found that both MERRA and NARR generally reproduce continental trends but exhibit significant differences at finer regional scales. These are a strength of this project for using our novel statistical techniques in climatology. Involving students and therefore helping them prepare for their careers are another important goal of this project. Over the project period, two graduate students, three undergraduate students, one high school student have participated in this project. One of the graduate students was directly supported by this grant for three regular semesters and one summer session. The three undergraduate students have received financial support from REU and STEP UG grants. Part of the research done by this project has been basis for the two graduate students's masters' degree theses. Their research experience greatly helped them for their job placements. The three undergraduate students presented their research outcomes in undergraduate STEM research conferences. The project outcomes have been also presented in 11 conferences and seminar meetings, including three Joint Statistical Meetings and three mathematics graduate students seminars at Boise State.

National Science Foundation (NSF)
Division of Mathematical Sciences (DMS)
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Gabor J. Szekely
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Boise State University
United States
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