Linear time series models

This chapter introduces the concepts of linear stochastic processes and linear time series models. The definition of linear stochastic processes is highly related to the theory of linear systems. The chapter briefly reviews the most important theory for linear systems. Linear systems are often most conveniently described by the transfer function, in the z-domain or in the s-domain for discrete time or continuous time systems, respectively. A linear stochastic process can be considered as generated from a linear system where the input is white noise. White noise is a sequence of uncorrelated, identically distributed random variables. Discrete time white noise is therefore sometimes referred to as a completely uncorrelated process or a pure random process. A very useful class of linear processes consists of those which have a rational transfer function. The AutoRegressive-Integrated-MovingAverage (ARIMA) process is very useful for describing some non-stationary behaviours like stochastic trends.