The first chapter addresses the degree to which models which exhibit nonlinear mean reversion, such as the smooth transition autoregressive model, present a resolution to the purchasing power parity puzzle (see Rogoff, 1996). A key contribution of this paper is the development of a method of estimating a representative distribution of half lives which is based upon the observed distribution of shocks in a given time series. This approach is implemented with data on four real exchange rates. The results suggest that while NMR may produce half lives lower than the three year benchmark, half lives shorter than two years are relatively uncommon. In the second chapter, the tests of Kapetanios, Shin, and Snell (2003) and Bec, Salem, and Carrasco (2004), which are designed to detect nonstationarity verses globally stationary ESTAR nonlinearity, are extended to allow transition variables with delay parameters greater than one. It is shown that both test statistics have the same asymptotic distribution compared with the case when the delay parameter is equal to one. The application of these generalized tests is illustrated in an empirical exercise using data on a set of 105 real exchange rates and 15 real interest rates. The third chapter investigates the small sample properties of threshold parameter estimation in the self exciting threshold autoregressive (SETAR) model. While it has been shown that the conditional least squares estimator in the SETAR model behaves poorly in general (see Kapetanios, 2000), this paper identifies systematic small sample biases that results when the distribution of observations between regimes is uneven. The importance of this issue is illustrated with Monte Carlo experiments based on estimating ?commodity points? in a law of one price framework. The fourth chapter combines approaches focusing on the role of distance and market heterogeneity to study what factors contribute to spatial violations of the law of one price. Using data on disaggregated cost of living indices for a cross section of 211 cities in the United States, evidence is provided that suggests that not controlling for market heterogeneity could produce misleading estimates of the role of distance on price differentials.