Models For Nonstationary Time Series

Any time series without a constant mean over time is nonstationary. Models of the form _Yt = µ _t + _Xt where µ _t is a nonconstant mean function and _Xt is a zero-mean, stationary series, were considered in Chapter 3. As stated there, such models are reasonable only if there are good reasons for believing that the deterministic trend is appropriate “forever.” That is, just because a segment of the series looks like it is increasing (or decreasing) approximately linearly, do we believe that the linearity is intrinsic to the process and will persist in the future? Frequently in applications, particularly in business and economics, we cannot legitimately assume a deterministic trend. Recall the random walk displayed in Exhibit 2.1, on page 14. The time series appears to have a strong upward trend that might be linear in time. However, also recall that the random walk process has a constant, zero mean and contains no deterministic trend at all.

As an example consider the monthly price of a barrel of crude oil from January 1986 through January 2006. Exhibit 5.1 displays the time series plot. The series displays considerable variation, especially since 2001, and a stationary model does not seem to be reasonable. We will discover in Chapters 6, 7, and 8 that no deterministic trend model works well for this series but one of the nonstationary models that have been described as containing stochastic trends does seem reasonable. This chapter discusses such models. Fortunately, as we shall see, many stochastic trends can be modeled with relatively few parameters.