Weighted Dickey–Fuller processes for detecting stationarity

Aiming at monitoring a time series to detect stationarity as soon as possible, we introduce monitoring procedures based on kernel-weighted sequential Dickey–Fuller (DF) processes, and related stopping times, which may be called weighted DF control charts. Under rather weak assumptions, (functional) central limit theorems are established under the unit root null hypothesis and local-to-unity alternatives. For general dependent and heterogeneous innovation sequences the limit processes depend on a nuisance parameter. In this case of practical interest, one can use estimated control limits obtained from the estimated asymptotic law. Another easy-to-use approach is to transform the DF processes to obtain limit laws which are invariant with respect to the nuisance parameter. We provide asymptotic theory for both approaches and compare their statistical behavior in finite samples by simulation.