New Unit Root Asymptotics in the Presence of Deterministic Trends

Recent work by the author (1998) has shown that stochastic trends can be validly represented in empirical regressions in terms of deterministic functions of time. These representations offer an alternative mechanism for modelling stochastic trends. It is shown here that the alternate representations affect the asymptotics of all commonly used unit root tests in the presence of trends. In particular, the critical values of unit root tests diverge when the number of deterministic regressors K approaches infinity as the sample size n approaches infinity. In such circumstances, use of conventional critical values based on fixed K will lead to rejection of the null of a unit root in favour of trend stationarity with probability one when the null is true. The results can be interpreted as saying that serious attempts to model trends by deterministic functions will always be successful and that these functions can validly represent stochastically trending data even when lagged variables are present in the regressor set, thereby undermining conventional unit root tests.