Statistical Analysis of Time Series with Long Memory

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During the last decades, long memory statistical models have become an important part of theoretical and applied time series analysis. These models are characterized by slowly decaying correlation functions at infinity, or by spectral densities possessing singularities (poles or zeros) at the origin. These features change in an essential way the statistical estimation and prediction procedures, and as a consequence, many of the methods and results used for analyzing short-memory time series models are no longer appropriate. The main objective of the proposal is to develop rigorous estimation and prediction procedures for a broad class of time series models that possess various types of memory structures. The PIs will study estimation and prediction problems for second order discrete or continuous time stationary random processes with spectral density functions that may have singularities, and the related analytical problems from Toeplitz operators theory, which serve as tools both in the discrete and continuous time case. In the estimation problem, the PIs will investigate the statistical properties of various estimators of the unknown parameters of the model, which depend on the memory structure and the smoothness of the spectral density. In the prediction problem the PIs will study the rate of decrease of the relative prediction error as the length of observed past increases. In many practical applications (for instance, economics, finance, computer science, hydrology), the data is well described by time series exhibiting both long memory and seasonality, or by some functions of such time series. The proposed model captures all these as special cases, and provides a sensible way to analyze certain macroeconomic time series (inflation and interest rates, monetary aggregates, revenue series, etc.), financial time series (volatility of financial asset returns, forward exchange market premia, etc.) as well as time series which arise in the analysis of computer traffic networks.