Properties of the ADF Unit Root Test for Models with Trends and Cycles.

A unit root test is usually carried out by using the regression test introduced by Dickey and Fuller (1979). Under the null hypothesis the series should be a random walk. But a non-stationary series can usually be decomposed into a random walk and a stationary component. This is what is done in additive decompositions between a trend and a cycle. The model considered here lies in the class of UC model developed by Harvey (1989) whose reduced form is an ARIMA (0,1,q) model. In this context, testing for a unit root can be compared to testing for a unit root in an ARIMA constrained model with a moving average polynomial. The paper analyzes the asymptotic distribution of the Dickey and Fuller (DF) test and of the Augmented Dickey and Fuller (ADF) test under this kind of null hypothesis.