Short, Intermediate, and Long Memory Properties of Time Series

This chapter provides a systematic classification of time series based on their memory characteristics, distinguishing between short, intermediate, and long memory processes. The classification framework utilizes the decay rate of autocorrelations and the behavior of the spectral density function: short memory processes exhibit exponential autocorrelation decay with bounded spectrum at the origin, while long memory processes display hyperbolic decay with unbounded spectral peaks. Intermediate memory processes occupy a transitional region between these extremes. The chapter presents a comprehensive evolution of long memory time series modeling, tracing developments from ARMA through ARFIMA to the generalized GARMA models. A pivotal contribution is the introduction of state space methodology for time series analysis, presenting the observation and transition equations that form the basis for Kalman filter applications. The recursive estimation framework is developed, explaining how parameters can be updated sequentially as new observations become available. The Kalman filter algorithm is introduced as a key tool for maximum likelihood estimation and prediction, enabling efficient computation of likelihood functions for long memory models. This chapter establishes the theoretical bridge between classical time series concepts and modern state space approaches that will be applied to long memory models in subsequent chapters.