FIGURE 21.9 First differences of U.S. GDP, 1970-1991 (quarterly).

In other words, run the following regression:

where Yt is the time series under study and where t is the trend variable measured chronologically. Now lit = (Y - Pi - P2t)

will be stationary. Ut is known as a (linearly) detrended time series.

It is important to note that the trend may be nonlinear. For example, it could be

which is a quadratic trend series. If that is the case, the residuals from (21.10.4) will now be (quadratically) detrended time series.

It should be pointed out that if a time series is DSP but we treat it as TSP, this is called underdifferencing. On the other hand, if a time series is TSP but we treat it as DSP, this is called overdifferencing. The consequences of these types of specification errors can be serious, depending on how one handles the serial correlation properties of the resulting error terms.38

In passing it may be noted that most macroeconomic time series are DSP rather than TSP.

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