This dissertation consists of six papers. Each of these papers are on a different aspect of statistical analysis of nonlinear time series. In the first paper, we study the behavior of a nonstationary time series which has different behavior for "high" and "low" levels. This consists of the introduction of a new nonlinear time series model, a mathematical analysis of the functional limit theorem for this model, a statistical test for behavior similar to this new model, and a proposed technique for robust cointegration in the presence of this new model. The second paper consists of an extension of this idea into volatility modeling. The third paper considers experimental design and sampling of Markov chains. In particular, it focuses on how to feasibly optimally sample a continuous two-state Markov chain. The fourth paper is on integer valued time series. The focus here is on studying the properties of the INGARCH(1,1) model in the nonstationary case. This consists of applying mathematical machinery rarely used in econometrics. Additionally, in this paper extensions towards stationarity tests are considered. The fifth paper studies the dynamic Tobit, a time series model often used when data is censored below. In this paper, weak dependence and mixing properties are shown to hold, which is relevant for studying the statistical properties of estimation for this model. The sixth paper studies the reciprocal of the random walk. This is relevant in time series econometrics as such a process is a possible model for time series with a stochastic diminishing trend.