This thesis discusses unconditional maximum likelihood estimation for testing for a unit root in seasonal autoregressive models. Chapter 1 is an introduction and chapter 2 develops the basic theory. The limiting distributions of the maximum likelihood estimators are shown and tables of the distributions of the estimators for several finite sample sizes and in the limit are given by Monte Carlo simulation. These can be used to test the hypothesis that a time series has a seasonal unit root. In chapters 3 and 4, the behavior of these unconditional maximum likelihood tests is studied for both nonseasonal explosive and seasonal explosive time series. The limiting distributions of the test statistics are derived and a certain unusual behavior of the limiting distributions is explained. A test is developed for the unit root null hypothesis against the explosive alternative, based on the unconditional maximum likelihood estimator. The results also make a two sided test against both the stationary and the explosive alternative possible. The unconditional maximum likelihood statistics including our pivotal statistics can be easily calculated using SAS. They are more powerful than the tests based on the least squares estimators in many models. In chapter 5, the limiting distributions of least squares estimators in seasonal time series models with a single trend are shown when the series has a seasonal unit root. The empirical distributions of the estimators are constructed for a seasonal unit root test. The least squares estimator has more powerful than unconditional maximum estimator when there is a single mean in seasonal models.
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