Time Series Analysis: Forecasting and Control, Fourth Edition

CHAPTER FOUR

Linear Nonstationary Models

Many empirical time series (e.g., stock prices) behave as though they had no fixed mean. Even so, they exhibit homogeneity in the sense that apart from local level, or perhaps local level and trend, one part of the series behaves much like any other part. Models that describe such homogeneous nonstationary behavior can be obtained by supposing some suitable difference of the process to be stationary. We now consider the properties of the important class of models for which the dth difference is a stationary mixed autoregressive–moving average process. These models are called autoregressive integrated moving average (ARIMA) processes.

4.1 AUTOREGRESSIVE INTEGRATED MOVING AVERAGE PROCESSES

4.1.1 Nonstationary First-Order Autoregressive Process

Figure 4.1 shows sections of four time series encountered in practice. These series have arisen in forecasting and control problems, and all of them exhibit behavior suggestive of nonstationarity. Series A, C, and D represent “uncontrolled” outputs (concentration, temperature, and viscosity, respectively) from three different chemical processes. These series were collected to show the effect on these outputs of uncontrolled and unmeasured disturbances such as variations in feedstock and ambient temperature. The temperature Series C was obtained by temporarily disconnecting the controllers on the pilot plant involved and recording the subsequent temperature fluctuations. Both A and D were collected on ...