New Unit Root Asymptotics in the Presence of Deterministric 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.
Phillips, Peter C.B., "New Unit Root Asymptotics in the Presence of Deterministric Trends" (1998). Cowles Foundation Discussion Papers. 1444.