2. Uncertainty about the unit root hypothesis
As argued by Christiano and Eichenbaum 1990, and more technically by Faust 1996, every DS process is arbitrarily close to some TS process, and vice versa. Cochrane 1991
has identified a similar observational equivalence between a TS process and a DS process when the variance of the innovation is small. Statistical distinction between such processes
is practically impossible, although it is practically important for inference, modeling, and forecasting at long-run horizons. However, given limited amounts of data, parsimoniously
parameterized TS and DS models achieved in practice can be quite distinct.
It is important to be precise about the null and alternative hypotheses of the traditionally used Dickey-Fuller test. In the present context, the null hypothesis is that a time series is
stationary after, but only after, first differencing. The alternative hypothesis is TS, generally about a linear trend, in levels. The emphasis on stationarity in differences under the null is
crucial, so that this hypothesis is best stated as DS. Rejection of the null is then rejection of DS.
3
Although the Dickey-Fuller test is designed to have power against TS, strong rejections of the null hypothesis can occur as a result of departures from DS other than TS. Similarly,
failure to reject the null of DS can be a manifestation of phenomena other than a DS data generating process. There are a number of possibilities.
4
First, Dickey-Fuller tests are based on the prior fitting of relatively low order autoregres- sions. It follows that these tests are valid only if the true generating process is a low order
autoregression, or can be adequately approximated by such. However, for some time series, the true generating process will incorporate a moving average component.
5
Such a compo- nent arises, for example, from simple models where an outcome is viewed as the sum of
permanent and transitory components. It is well known Schwert, 1989; Agiakloglou and Newbold, 1992 that large moving average components generate spurious rejections by
Dickey-Fuller tests. Ng and Perron 1995 prove that use of particular criteria for selecting autoregressive order in Dickey-Fuller tests leave the asymptotic null distribution undisturbed
for a range of generating models involving moving average terms.
6
While this seems about the best that can be done, Agiakloglou and Newbold 1996 demonstrate for typical sample sizes,
that although the spurious rejection problem is somewhat alleviated, it may be far from cured. Second, since the seminal work of Perron 1989, it has been well-known that structural
breaks in a time series can severely reduce the power of Dickey-Fuller tests. Perron 1989 illustrated that the usual Dickey-Fuller tests of the unit root hypothesis can have low power
when the true data generating process DGP is stationary about a broken linear trend. In these cases the standard Dickey-Fuller tests will fail to find strong evidence against the null
of unit root. Perron 1989 incorporating dummy variables at possible break points in the series and allowing for an exogenous level shift in the trend, found that he could reject the
null of a stochastic trend for RGNP over the period 1909 –1970 in favor of the alternative of deterministic trend with a shift in the level of the trend.
7
Zivot and Andrews 1992, applying a sequential Dickey-Fuller test endogenizing the break point and adjusted critical values to
the Nelson-Plosser data, find slightly less evidence against the null hypothesis of unit root null than Perron 1989. Balke and Fomby 1991 have argued that a series subjected to
infrequent permanent shocks is indistinguishable from a series with a trend break. In a recent paper Kilian and Ohanian 1998 present Monte Carlo simulations that show that a DGP that
87 P. Newbold et al. Journal of Economics and Business 53 2001 85–102
is the sum of an integrated component with drift and an occasional large transitory compo- nent will generate data in finite samples that is difficult to distinguish from a trend stationary
process with a trend break. They report that purely transitory disturbances can cause rejection of the unit root null hypothesis and spurious results indicating that there has been a
permanent shift in the level.
Third, it is also the case that outliers Franses and Haldrup, 1994 or structural breaks Leybourne, Mills and Newbold, 1998 can induce spurious rejections by Dickey-Fuller
tests. This issue was raised by Nelson and Murray 2000 in discussing tests for unit roots in RGNP. Lengthening the span of data increases the possibility of major structural changes.
Failure to account for such structural changes will bias unit root tests. For example, Leybourne, Mills and Newbold 1998 found results converse to those of Perron 1989.
They show that when a series is generated by a process that is I1, with an early but not late break, application of the Dickey-Fuller test will lead to spurious rejection of the unit root null
hypothesis. These results suggest that the application of the Dickey-Fuller tests will yield misleading results when applied to data that have a structural break early in the series. One
important implication of these findings is that studies that justify using longer spans of data on the grounds of increasing the power of the tests and more precise estimates, can lead to
erroneous findings if the early data contains a structural break.
In our view, statistical tests of unit roots will almost inevitably be based explicitly or implicitly on models that are drastic simplifications of the underlying generating process. It
is simply impossible, given available data, to simultaneously contemplate models that are more highly parameterized than the typical parsimonious forms, including possible moving
average terms, and allow for the possibility of both outliers and structural breaks of unknown form and type, occurring at unknown points in time. Whether the simplified structures on
which the tests are necessarily based are helpful or misleading is difficult to anticipate in any application.
3. US real GNP, 1875–1993
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