The starting point for the analysis of series of observations indexed by time {yt, t ≥ 0} is to examine their graphical representations. Oftentimes it happens that the series exhibit an explosive pattern, that is, they give the impression of tending toward infinity with t. In such a case, the attention is focused on the dominant components of the series which are smoother than the original ones, but asymptotically equivalent to them. In this chapter we will mainly stress the importance of such components, which we will assume are diverging in a polynomial fashion. It is possible to obtain such a behavior through a nonlinear transformation of the original series most of the time. We can ask a number of questions about this trend component, according to whether we examine the series separately or jointly.

What is the rate of divergence of the various series? What are the differences among them?

What happens to the series once the dominant component is eliminated; are there still some diverging components and how important are they?

The joint plot of two series sometimes shows fairly strong links among the trend components of the series. Is it possible to make these links explicit, to study the cases where they are particularly strong, and to compare the strength of these links with those of other components of the series?