Bounded integrated processes and unit root tests

In the framework of integrated processes, the problem of testing the presence of unknown boundaries which constrain the process to move within a closed interval is considered. To analyze this problem, the concept of bounded integrated process is introduced, thus allowing to formally define boundary conditions for I(1) processes. A new class of tests, which are based on the rescaled range of the process, is introduced in order to test the null hypothesis of no boundary conditions. The limit distribution of the test statistics involved can be expressed in terms of the distribution of the range of Brownian functionals, while the power properties are obtained by deriving some asymptotic results for I(1) processes with boundary conditions. Both theoretical and simulation investigations show that range-based tests outperform standard unit root tests significantly when used to detect the presence of boundary conditions. A previous draft of the paper (Cavaliere, 2000) was presented at the 8th World Congress of the Econometric Society, Seattle, 11–16 August 2000. I wish sincerely to thank: Martin Jacobsen for his patience in discussing weak convergence to regulated Brownian motions and his valuable suggestions; the Department of Theoretical Statistics of the University of Copenhagen whose hospitality is gratefully acknowledged; Tommaso Proietti for important suggestions; Silvano Bordignon and partecipants at the CIdE seminar, University of Padua, June 2000; two anonymous referees. Partial financial support from 60% M.U.R.S.T. research grants is acknowledged.