Stochastic unit root processes: Maximum likelihood estimation, and new Lagrange multiplier and likelihood ratio tests

We show in this study that the maximum likelihood estimators of stochastic unit root (STUR) processes are consistent and asymptotically normally distributed. We also present two new tests for STUR. We first propose a Lagrange multiplier test and show that it has a standard χ2 distribution asymptotically. We also propose a likelihood ratio test and show that it has an asymptotic distribution of 50-50 mixture of χ2 and a point mass at 0. As an empirical example, we test the existence of STUR in the Canadian real exchange rate and explore the implication of STUR on the validity of purchasing power parity.