ABSTRACT
In Chapters 3 and 4, we considered stationary processes. However, many or even most processes of interest tend to have some type of nonstationary or near-nonstationary behavior. Referring to the definition of stationarity in Definition 1.5, a process can be nonstationary by having (1) a nonconstant mean, (2) a nonconstant or infinite variance, or (3) an autocorrelation function that depends on time. The processes we discuss in this chapter will have one or more of these properties. In Chapters 12 and 13, we discuss techniques for analyzing time series having property (3). In this chapter, we will discuss the following:
. Deterministic signal-plus-noise models
. ARIMA, ARUMA, and nonstationary multiplicative seasonal models
. Random walk models with and without drift
. Time-varying frequency (TVF) models (or time-varying autocorrelation [TVA] models)
In this section, we consider models of the form
Xt ¼ st þ Zt, (5:1)
where st is a deterministic signal and Zt is a zero-mean stationary noise component that may or may not be uncorrelated. In this book, we will model Zt using an AR(p) model although other
models (e.g., ARMA(p,q)) could be used. We consider two types of signals: (a) ‘‘trend’’ signals, such as st ¼ aþ bt or st ¼ aþ b1tþ b2t2, . . ., and (b) harmonic signals, such as st ¼ C cos (2pftþU). We emphasize that signal-plusnoise models are quite restrictive, and they should only be considered if there are physical reasons to believe that a deterministic signal is present.
