Simple Tests for Short Memory in ARFIMA Models

Publication Date


Type of Culminating Activity


Degree Title

Master of Science in Mathematics



Major Advisor

Jaechoul Lee


For an autoregressive fractionally integrated moving-average ARFIMA(p, d, q) process, it is often a priori known that the process involves long memory. The objective of this thesis is to test whether the long memory time series needs other short memory parameters in the fractional model. To do this, the asymptotic distribution of sample autocorrelations is further developed using recognized and well-known theoretical results. Simple functions of sample autocorrelations are considered in a way that they satisfy √ n-consistency and their limiting distributions exhibit a normal distribution regardless of the true differencing parameter d. The advantages of using sample autocorrelations for testing autoregressive moving-average parameters lie in ease of the computation and use of the established asymptotic theories. While it is not the main point, we also provide a brief review of ARFIMA(p, d, q) processes and lay the foundation for a new technique to estimate the differencing parameter d. Additionally, the highly regarded Whittle estimate for the parameter d is further explored and utilized in this process. New graphical identification methods for short memory are also presented. These proposed methods are then applied to well-known long memory time series data in finance and hydrology.

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