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696 Chapter 16 Additional Topics in Time Series Regression
Key Concept Orders of Integration, Differencing, and Stationarity
16.4 • If Yt is integrated of order one—that is, if Yt is I(1)—then Yt has a unit autoregressive
root and its first difference, ∆Yt, is stationary.
• If Yt is integrated of order two—that is, if Yt is I(2)—then ∆Yt has a unit
autoregressive root and its second difference, ∆2Yt, is stationary.
• If Yt is integrated of order d—that is, if Yt is I(d )—then Yt must be differ-
enced d times to eliminate its stochastic trend; that is, ∆dYt is stationary.
difference is stationary. If a series has a trend of the form in Equation (16.17), then
the first difference of the series has an autoregressive root that equals 1.
“Orders of integration” terminology. Some additional terminology is useful for
distinguishing between these two models of trends. A series that has a random
walk trend is said to be integrated of order one, or I(1). A series that has a trend
of the form in Equation (16.17) is said to be integrated of order two, or I(2). A
series that does not have a stochastic trend and is stationary is said to be inte-
grated of order zero, or I(0).
The order of integration in the I(1) and I(2) terminology is the number of
times that the series needs to be differenced for it to be stationary: If Yt is I(1),
then the first difference of Yt, ∆Yt, is stationary, and if Yt is I(2), then the second
difference of Yt, ∆2Yt, is stationary. If Yt is I(0), then Yt is stationary.
Orders of integration are summarized in Key Concept 16.4.
How to test whether a series is I(2) or I(1). If Yt is I(2), then ∆Yt is I(1), so ∆Yt has
an autoregressive root that equals 1. If, however, Yt is I(1), then ∆Yt is stationary.
Thus the null hypothesis that Yt is I(2) can be tested against the alternative
hypothesis that Yt is I(1) by testing whether ∆Yt has a unit autoregressive root. If
the hypothesis that ∆Yt has a unit autoregressive root is rejected, then the hypoth-
esis that Yt is I(2) is rejected in favor of the alternative that Yt is I(1).
Examples of I(2) and I(1) series: The price level and the rate of inflation. The rate
of inflation is the growth rate of the price level. Recall from Section 14.2 that the
growth rate of a time series Xt can be computed as the first difference of the loga-
rithm of Xt; that is Δln(Xt) is the growth rate of Xt (expressed as fraction). If Pt is
