Testing for unit roots in the presence of uncertainty over both the trend and initial condition

In this paper we provide a joint treatment of two major problems that surround testing for a unit root in practice: uncertainty as to whether or not a linear deterministic trend is present in the data, and uncertainty as to whether the initial condition of the process is (asymptotically) negligible or not. We suggest decision rules based on the union of rejections of four standard unit root tests (OLS and quasi-differenced demeaned and detrended ADF unit root tests), along with information regarding the magnitude of the trend and initial condition, to allow simultaneously for both trend and initial condition uncertainty.