Estimation, testing, and forecasting for long memory processes

Chuanbo Zang, Purdue University

Abstract

Empirical experience over the last few decades has shown that standard iid theory or short range dependence time series models do not adequately describe many types of real data in various areas of applications such as geology, astronomy, economics, hydrology, etc. Time series models with long memory or long range dependence have been introduced and applied quite successfully to analyze such data. The area has been growing at a fast rate, with new methods emerging consistently. The theoretical literature has been primarily frequentist and asymptotic. The corresponding theoretical studies in a Bayesian framework had been lacking. In this work, we study parameter estimation, construction of noninformative priors, and forecasts of future observations in a Bayesian decision theoretic setup. We show that for point estimation of the mean of a Gaussian Fractional ARIMA process, simple estimates such as the sample mean suffice, but for interval estimation and point estimation of scale, long memory has peculiar and even calamitous effects on the performance of simple estimates. At a more basic level, just the presence of long memory can have dramatic effects on the quality of general inference and on basic Bayesian tools such as Bayes factors and their asymptotic behavior. Specifically, the null asymptotic distributions of Bayes factors and their rates of convergence depend on the value of the long memory parameter. Long memory also results in catastrophic sample size requirements just to have accuracies comparable to the iid case. Comparative evaluations show that Bayesian methods perform demonstrably better than certain other methods including the empirical BLUP and the MLE in a number of problems. There seems to be a real potential for success of Bayesian methods in other types of long memory processes.

Degree

Ph.D.

Advisors

DasGupta, Purdue University.

Subject Area

Statistics

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