On detection of unit roots generalizing the classic Dickey-Fuller approach

If we are given a time series of economic data, a basic question is whether the series is stationary or a random walk, i.e., has a unit root. Whereas the problem to test the unit root null hypothesis against the alternative of stationarity is well studied in the context of classic hypothesis testing in the sense of Neyman, sequential and monitoring approaches have not been studied in detail yet. We consider stopping rules based on a sequential version of the well known Dickey-Fuller test statistics in a setting, where the asymptotic distribution theory becomes a nice and simple application of weak convergence of Ito integrals. More sophisticated extensions studied elsewhere are outlined. Finally, we present a couple of simulations.