Time Series Analysis: Forecasting and Control, Fourth Edition

PART ONE

Stochastic Models and Their Forecasting

In the first part of this book, which includes Chapters 2, 3, 4, and 5, a valuable class of stochastic models is described and its use in forecasting discussed.

A model that describes the probability structure of a sequence of observations is called a stochastic process. A time series of N successive observations z′ = (z₁, z₂, …, z_N is regarded as a sample realization, from an infinite population of such samples, which could have been generated by the process. A major objective of statistical investigation is to infer properties of the population from those of the sample. For example, to make a forecast is to infer the probability distribution of a future observation from the population, given a sample z of past values. To do this we need ways of describing stochastic processes and time series, and we also need classes of stochastic models that are capable of describing practically occurring situations. An important class of stochastic processes discussed in Chapter 2 is the stationary processes. They are assumed to be in a specific form of statistical equilibrium, and in particular, vary over time in a stable manner about a fixed mean. Useful devices for describing the behavior of stationary processes are the autocorrelation function and the spectrum.

Particular stationary stochastic processes of value in modeling time series are the autoregressive, moving average, and mixed autoregressive–moving average processes. The properties ...