Transforming Nonstationary Time Series

Now that we know the problems associated with nonstationary time series, the practical question is what to do. To avoid the spurious regression problem that may arise from regressing a nonstationary time series on one or more nonstationary time series, we have to transform nonstationary time series to make them stationary. The transformation method depends on whether the time series are difference stationary (DSP) or trend stationary (TSP). We consider each of these methods in turn.

Difference-Stationary Processes

If a time series has a unit root, the first differences of such time series are stationary.37 Therefore, the solution here is to take the first differences of the time series.

Returning to our U.S. GDP time series, we have already seen that it has a unit root. Let us now see what happens if we take the first differences of the GDP series.

Let AGDPt = (GDPt — GDPt—1). For convenience, let Dt = AGDPt. Now consider the following regression:

The 1 percent critical DF t value is —3.5073. Since the computed t (= t) is more negative than the critical value, we conclude that the first-differenced GDP is stationary; that is, it is 7(0). It is as shown in Figure 21.9. If you compare Figure 21.9 with Figure 21.1, you will see the obvious difference between the two.

Trend-Stationary Process

As we have seen in Figure 21.5, a TSP is stationary around the trend line. Hence, the simplest way to make such a time series stationary is to regress it on time and the residuals from this regression will then be stationary.

37If a time series is 7(2), it will contain two unit roots, in which case we will have to difference it twice. If it is 7(d), it has to be differenced d times, where d is any integer.


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