In this paper we will attempt to apply a new test, developed, on several time series, such as stock indexes and exchange rates, a test that is characterized by an even general setting. Moreover, it extends the theory developed by Phillips to include cases where the variance grows without limit in a polynomial fashion. In particular they relax the restrictive assumption that are all bounded, that is for some β>2, thus precluding trending moments. What is interesting about this test is that it embodies the Phillips-Perron traditional test as a special case. The discussion that follows is organized in four parts: In the first part, we analyze the probabilistic structure of autoregressive models, and particular those of order one, which host the random walk models as a special case. In the second part, we examine some of the important properties of random walk models and we display the difference between them and the stable autoregressive ones. The third part is divided into two parts: In the first part we give an explicit citation of unit root tests; especially the Phillips-Perron framework is given a special treat and in the second part the new test is presented in detail. Finally, in the fourth path we present the methodology that was used in the implementation of the test and the results from it. The details about the data and the rundown of the results can be found in the appendix.