Page 696 -

16.3 Orders of Integration and the DF-GLS Unit Root Test 695

in multivariate forecasting models with many predictors, in which case a VAR
specified in terms of all the variables could be unreliable because it would have
very many estimated coefficients.

16.3 Orders of Integration and the DF-GLS

Unit Root Test

This section extends the treatment of stochastic trends in Section 14.6 by address-
ing two further topics. First, the trends of some time series are not well described
by the random walk model, so we introduce an extension of that model and dis-
cuss its implications for regression modeling of such series. Second, we continue
the discussion of testing for a unit root in time series data and, among other things,
introduce a second test for a unit root, the DF-GLS test.

Other Models of Trends and Orders of Integration

Recall that the random walk model for a trend, introduced in Section 14.6, speci-
fies that the trend at date t equals the trend at date t - 1, plus a random error
term. If Yt follows a random walk with drift b0, then

Yt = b0 + Yt - 1 + ut, (16.16)

where ut is serially uncorrelated. Also recall from Section 14.6 that, if a series has
a random walk trend, then it has an autoregressive root that equals 1.

Although the random walk model of a trend describes the long-run move-
ments of many economic time series, some economic time series have trends that
are smoother—that is, vary less from one period to the next—than is implied by
Equation (16.16). A different model is needed to describe the trends of such
series.

One model of a smooth trend makes the first difference of the trend follow a
random walk—that is,

∆Yt = b0 + ∆Yt - 1 + ut, (16.17)

where ut is serially uncorrelated. Thus, if Yt follows Equation (16.17), ∆Yt follows a
random walk, so ∆Yt - ∆Yt - 1 is stationary. The difference of the first differences,
∆Yt - ∆Yt - 1, is called the second difference of Yt and is denoted ∆2Yt =
∆Yt - ∆Yt - 1. In this terminology, if Yt follows Equation (16.17), then its second