Abstract :
By pointing out the spurious regression problem, Granger and Newbold ~1974!
have shown the importance of stochastic trends in time series data in the context
of linear regression models+ At the time, removing trends by differencing
was already common practice in univariate time series modeling as part of the
Box–Jenkins approach ~Box and Jenkins, 1976!+ These new developments, however,
emphasized the importance of autoregressive ~AR! unit roots and motivated
the development of statistical procedures for their detection+ Dickey and
Fuller ~1979! and Fuller ~1976! were pioneers in developing tests for unit roots
that became widely used+ The foundation of asymptotic theory for regressions
involving stochastic trends was led by Phillips ~1986, 1987! with the introduction
of the functional limit theory, weak convergence methods, convergence to
stochastic integrals, nonparametric unit root testing, and continuous record
asymptotics+ Phillips and Durlauf ~1986! extended some of these results to the
multivariate setting by presenting the multivariate invariance principles and the
asymptotic theory of multivariate nonstationary and cointegrating regressions+
These contributions provided the asymptotic tools that have served as the basis
for most of the limit results derived in the context of unit root nonstationarity,
and they have stimulated extensive subsequent research+
Over the last 30 years many other important contributions were made, of which
the following are of particular relevance+ Augmented Dickey–Fuller ~ADF! tests
