Time Series Analysis: Forecasting and Control, Fourth Edition

CHAPTER TWO

Autocorrelation Function and Spectrum of Stationary Processes

A central feature in the development of time series models is an assumption of some form of statistical equilibrium. A particular assumption of this kind (an unduly restrictive one, as we shall see later) is that of stationarity. Usually, a stationary time series can be usefully described by its mean, variance, and autocorrelation function, or equivalently by its mean, variance, and spectral density function. In this chapter we consider the properties of these functions and, in particular, the properties of the autocorrelation function, which is used extensively in the chapters that follow.

2.1 AUTOCORRELATION PROPERTIES OF STATIONARY MODELS

2.1.1 Time Series and Stochastic Processes

Time Series

A time series is a set of observations generated sequentially in time. If the set is continuous, the time series is said to be continuous. If the set is discrete, the time series is said to be discrete. Thus, the observations from a discrete time series made at times τ₁, τ₂, …, τ_t, …, τ_N may be denoted by z(τ₁), z(τ₂), …, z(τ_t), …, z(τ_N). In this book we consider only discrete time series where observations are made at a fixed interval h. When we have N successive values of such a series available for analysis, we write z₁, z₂, …, z_t, …, z_N to denote observations made at equidistant time intervals τ₀ + h, τ₀ + 2h, …, τ₀ + th, …, τ₀ + Nh. For many purposes the values of τ₀ and h are unimportant, but if the observation ...