Unit-root test within a threshold ARMA framework

We propose a new unit-root test based on Lagrange Multipliers, where we extend the null hypothesis to an integrated moving-average process (IMA(1,1)) and the alternative to a first-order threshold autoregressive moving-average process (TARMA(1,1)). This new theoretical framework provides tests with good size without pre-modelling steps. Moreover, leveraging on the versatile capability of the TARMA(1,1), our test has power against a wide range of linear and nonlinear alternatives. We prove the consistency and asymptotic similarity of the test. The proof of tightness of the test is of independent and general theoretical interest. Moreover, we propose a wild bootstrap version of the statistic. Our proposals outperform most existing tests in many contexts. We support the view that rejection does not necessarily imply nonlinearity so that unit-root tests should not be used uncritically to select a model. Finally, we present an application to real exchange rates.