Traditional linear cointegration models have been widely used to examine longrun relationships between economic variables; however, empirical evidence suggests that the linear structure fails to account for economic changes due to technology improvement, business cycles and policy alterations. Nonlinear cointegration models provide an important means to extend conventional cointegration analysis by incorporating these factors. In the first chapter, I establish a statistical theory for cointegrating regressions with threshold effects. I derive asymptotics of the profiled least square (LS) estimators assuming the size of the threshold effect converges to zero. Depending on how rapidly this sequence converges, the model may be identified or weakly identified. A model-selection procedure is then applied to construct robust confidence intervals, which have approximately correct coverage probability irrespective of the magnitude of the threshold effect. Using a parametric model, however, one always suffers from the danger of model misspecifications. The standard tests based on parametric models cannot tell us whether the rejection or acceptance of threshold effects is due to real regime shifts or a functional misspecification. In the second chapter, I consider the estimation and testing for threshold effects in regression models with unknown functional forms. I use series expansions to approximate the unknown regression functions and estimate the threshold effect with a profile least square method. A nice property of the estimator is that it achieves T-convergence rate as in parametric models. I derive the asymptotic distribution of the threshold estimator and design a generalized sup Wald statistic to test the threshold effect. In the third chapter, I consider an application of threshold cointegration on the price discovery for cross-listed stocks. For cross-listings, the convergence to equilibrium parity between home and guest market prices could be discontinuous, i.e., convergence may be quicker when the price deviation is sufficiently profitable. By considering the concept of threshold cointegration, I modify Harris et al.'s (1995, 2002) common factor approach to estimate the relative extent of market-respective contribution to price discovery. The method is applied to Canadian stocks cross-listed on the New York Stock Exchange (NYSE) and the Toronto Stock Exchange.