A Rescaled Range Statistics Approach to Unit Root Tests

In the framework of integrated processes, the problem of testing the presence of unknown boundaries which constrain the sample path to lie within a closed interval is considered. To discuss this inferential problem, the concept of nearly-bounded integrated process is introduced, thus allowing to define formally the concept of boundary conditions within I(1) processes. When used to detect unknown boundaries, standard unit root tests do not maintain the usual power properties and new methods need developing. Therefore a new class of tests, which are based on the rescaled range of the process, are introduced. The limiting distribution of the proposed tests can be expressed in terms of the distribution of the range of particular Brownian functionals, while the power properties are obtained through the derivation of the limiting Brownian functional of a I(1) process with boundary conditions, which is done by referring to a new invariance principles for nonstationary time series with limited sample paths. Both theoretical and simulation exercises show that range-based tests outperform standard unit root tests significantly when used to detect the presence of boundary conditions.