Threshold Autoregressive Moving-Average models: probabilistic structure, statistical aspects and applications

The thesis analyses threshold autoregressive moving-average models (TARMA). They are an extension of threshold autoregressive models (TAR) as to allow serially dependent noise. TARMA models describe parsimoniously many non-linear phenomena. The systematic study of TARMA models presents several challenges. The thesis solves the probabilistic problems for the first order TARMA models enabling their practical application. The results allow to develop a powerful unit root test for both linear and non-linear processes. Most unit root tests are affected by size distortion in presence of dependent errors, especially of the moving-average kind. The proposals that address such problem do not consider non-linear alternatives. On the other hand, tests that have a non-linear specification in the alternative hypothesis do not deal with the size issue. We use TARMA models to develop a novel unit root test based upon Lagrange multipliers that does not suffer from size distortions and, at the same time, allows for a wide and flexible non-linear alternative. We prove that our supLM test is consistent, it is similar and it is nuisance-parameters free. In addition to the asymptotic version of the test we propose a wild bootstrap version with very good properties in terms of size and power. The final part of the thesis is devoted to a preliminary empirical investigation regarding the parsimony of TARMA models. Moreover, we use TARMA models to analyse the Canadian lynx time series.